How many questions from Sequence and Series appear in JEE?

Sequence and Series has appeared in 164 of the last 186 JEE Main and JEE Advanced papers — about 88% of them — contributing 287 questions in total across those papers.

Is Sequence and Series an important chapter for JEE?

Judged by how often it is actually tested, it appears in roughly 88% of papers. Chapters above about 50% are effectively guaranteed to show up every session, so they repay thorough preparation; lower-frequency chapters are better treated as targeted revision.

Where do these Sequence and Series questions come from?

Every question is from an official JEE Main or JEE Advanced paper, transcribed from the original paper and tagged to this chapter. Answers follow the official answer key.

Sequence and Series JEE Previous Year Questions with Answers

Q1·MathematicsSingle correctJEE Main 2026

Let

\alpha = 3 + 4 + 8 + 9 + 13 + 14 + \ldots

upto 40 terms. If

(\tan \beta)^{\frac{\alpha}{1020}}

is a root of the equation

x^2 + x - 2 = 0

,

\beta \in \left(0, \frac{\pi}{2}\right)

, then

\sin^2 \beta + 3\cos^2 \beta

is equal to:

(A)2
(B) $\frac{7}{4}$
(C) $\frac{5}{2}$
(D) $\frac{3}{2}$

Q2·MathematicsSingle correctJEE Main 2026

The value of 1

^3

− 2

^3

+ 3

^3

− ... + 15

^3

is:

(A)1706
(B)1856
(C)1982
(D)2403

Q3·MathematicsNumericalJEE Main 2026

For the functions f(θ) = α tan

^2

θ + β cot

^2

θ, and g(θ) = α sin

^2

θ + β cos

^2

θ, α > β > 0, let

\min_{0<\theta<\pi/2} f(\theta) = \max_{0<\theta<\pi} g(\theta)

. If the first term of a G.P. is

\left(\frac{\alpha}{2\beta}\right)

, its common ratio is

\left(\frac{2\beta}{\alpha}\right)

and the sum of its first 10 terms is

\frac{m}{n}

, gcd(m, n) = 1, then m + n is equal to ______.

Q4·MathematicsSingle correctJEE Main 2026

The sum

1 + \frac{1}{2}(1^{2} + 2^{2}) + \frac{1}{3}(1^{2} + 2^{2} + 3^{2}) + \ldots

upto 10 terms is equal to :

(A)130
(B)155
(C) $\frac{315}{2}$
(D) $\frac{325}{2}$

Q5·MathematicsSingle correctJEE Main 2026

Consider the quadratic equation

(n^{2} - 2n + 2)x^{2} - 3x + (n^{2} - 2n + 2)^{2} = 0

,

n \in \mathbb{R}

. Let α be the minimum value of the product of its roots and β be the maximum value of the sum of its roots. Then the sum of the first six terms of the G.P., whose first term is α and the common ratio is

\frac{\alpha}{\beta}

, is :

(A) $\frac{61}{37}$
(B) $\frac{121}{81}$
(C) $\frac{364}{243}$
(D) $\frac{1093}{729}$

Q6·MathematicsSingle correctJEE Main 2026

The sum of the first ten terms of an A.P. is 160 and the sum of the first two terms of a G.P. is 8. If the first term of the A.P. is equal to the common ratio of the G.P. and the first term of the G.P. is equal to common difference of the A.P., then the sum of all possible values of the first term of the G.P. is:

(A) $\frac{34}{9}$
(B) $\frac{34}{13}$
(C) $\frac{32}{9}$
(D) $\frac{32}{13}$

Q7·MathematicsSingle correctJEE Main 2026

\sum_{n=1}^{10}\left(\frac{528}{n(n+1)(n+2)}\right)

is equal to:

(A)65
(B)130
(C)220
(D)440

Q8·MathematicsSingle correctJEE Main 2026

If the sum of the first 10 terms of the series

\frac{1}{1 + 1^4 \times 4} + \frac{2}{1 + 2^4 \times 4} + \frac{3}{1 + 3^4 \times 4} + \frac{4}{1 + 4^4 \times 4} + \ldots

is

\frac{m}{n}

, gcd(m, n) = 1, then

m + n

is equal to :

(A)256
(B)264
(C)276
(D)284

Q9·MathematicsSingle correctJEE Main 2026

Let

A_1, A_2, A_3, \ldots, A_{39}

be 39 arithmetic means between the numbers 59 and 159. Then the mean of

A_{25}, A_{28}, A_{31}

and

A_{36}

is equal to :

(A)129
(B)136
(C)131.50
(D)134

Q10·MathematicsSingle correctJEE Main 2026

Let the sum of the first

n

terms of an A.P. be

3n^2 + 5n

. Then the sum of squares of the first 10 terms of the A.P. is:

(A)10220
(B)12860
(C)15220
(D)19780

Q11·MathematicsSingle correctJEE Main 2026

The first term of an A.P. of 30 non-negative terms is

\frac{10}{3}

. If the sum of this A.P. is the cube of its last term, then its common difference is:

(A) $\frac{5}{87}$
(B) $\frac{25}{83}$
(C) $\frac{15}{29}$
(D) $\frac{5}{29}$

Q12·MathematicsSingle correctJEE Main 2026

Let

\alpha = \frac{1}{4} + \frac{1}{8} + \frac{1}{16} + \ldots \infty

and

\beta = \frac{1}{3} + \frac{1}{9} + \frac{1}{27} + \ldots \infty

. Then the value of

(0.2)^{\log_{\sqrt{5}}(\alpha)} + (0.04)^{\log_{5}(\beta)}

is equal to:

(A)4
(B)5
(C)8
(D)25

Q13·MathematicsSingle correctJEE Main 2026

Let

a_1, a_2, a_3, \ldots

be an A.P. and

g_1 = a_1, g_2, g_3, \ldots

be an increasing G.P. If

a_1 = a_2 + g_2 = 1

and

a_3 + g_3 = 4

, then

a_{10} + g_5

is equal to:

(A) $81$
(B) $76$
(C) $62$
(D) $55$

Q14·MathematicsSingle correctJEE Main 2026

Let A be the set of first 101 terms of an A.P., whose first term is 1 and the common difference is 5 and let B be the set of first 71 terms of an A.P., whose first term is 9 and the common difference is 7. Then the number of elements in A ∩ B, which are divisible by 3, is :

(A)4
(B)5
(C)6
(D)7

Q15·MathematicsNumericalJEE Main 2026

If

\sum_{k=1}^{n} a_{k} = 6n^{3}

, then

\sum_{k=1}^{6}\left(\frac{a_{k+1} - a_{k}}{36}\right)^{2}

is equal to ________.

Q16·MathematicsSingle correctJEE Main 2026

The sum

\frac{1^3}{1} + \frac{1^3 + 2^3}{1 + 3} + \frac{1^3 + 2^3 + 3^3}{1 + 3 + 5} + \cdots

up to 8 terms, is:

(A) $70$
(B) $71$
(C) $72$
(D) $73$

Q17·MathematicsSingle correctJEE Main 2026

The common difference of the A.P.:

a_1

,

a_2

, ….,

a_m

is 13 more than the common difference of the A.P.:

b_1

,

b_2

, ….,

b_n

. If

b_{31}

= –277,

b_{43}

= –385 and

a_{78}

= 327, then

a_1

is equal to

(A)21
(B)24
(C)19
(D)16

Q18·MathematicsSingle correctJEE Main 2026

The value of

\sum\limits_{k=1}^{\infty} (-1)^{k+1}\left(\dfrac{k(k+1)}{k!}\right)

is :

(A) $2/e$
(B) $1/e$
(C) $\sqrt{e}$
(D) $e/2$

Q19·MathematicsNumericalJEE Main 2026

In a G.P., if the product of the first three terms is 27 and the set of all possible values for the sum of its first three terms is

\mathbb{R}

– (a, b ), then

a^{2} + b^{2}

is equal to ____.

Q20·MathematicsNumericalJEE Main 2026

If

\sum_{r=1}^{25}\left(\frac{r}{r^4 + r^2 + 1}\right) = \frac{p}{q}

, where p and q are positive integers such that gcd (p, q) = 1, then p + q is equal to ______.

Q21·MathematicsSingle correctJEE Main 2026

\frac{6}{3^{26}} + \frac{10.1}{3^{25}} + \frac{10.2}{3^{24}} + \frac{10.2^2}{3^{23}} + ...+ \frac{10.2^{24}}{3}

is equal to

(A) $2^{25}$
(B) $2^{26}$
(C) $3^{25}$
(D) $3^{26}$

Q22·MathematicsSingle correctJEE Main 2026

Let 729, 81, 9, 1, …. be a sequence and

P_{n}

denote the product of the first n terms of this sequence. If

2\sum_{n=1}^{40}\left(P_{n}\right)^{\frac{1}{n}}=\frac{3^{\alpha}-1}{3^{\beta}}

and gcd (α, β) = 1, then α + β is equal to

(A) $73$
(B) $74$
(C) $75$
(D) $76$

Q23·MathematicsSingle correctJEE Main 2026

Consider an A.P.:

a_{1}, a_{2},....,a_{n}

;

a_{1} > 0

. If

a_{2} - a_{1} = \frac{-3}{4}

,

a_{n} = \frac{1}{4} a_{1}

, and

\sum\limits_{i=1}^{n} a_{i} = \frac{525}{2}

, then

\sum\limits_{i=1}^{17} a_{i}

is equal to

(A)476
(B)952
(C)238
(D)136

Q24·MathematicsSingle correctJEE Main 2026

\left(\frac{1}{3} + \frac{4}{7}\right) + \left(\frac{1}{3^{2}} + \frac{1}{3} \times \frac{4}{7} + \frac{4^{2}}{7^{2}}\right) + \left(\frac{1}{3^{3}} + \frac{1}{3^{2}} \times \frac{4}{7} + \frac{1}{3} \times \frac{4^{2}}{7^{2}} + \frac{4^{3}}{7^{3}}\right)

+ ...... upto infinite terms is equal to -

(A) $\frac{5}{2}$
(B) $\frac{7}{4}$
(C) $\frac{4}{3}$
(D) $\frac{6}{5}$

Q25·MathematicsSingle correctJEE Main 2026

Let

\sum_{k=1}^{n} a_k = \alpha n^2 + \beta n

. If

a_{10} = 59

and

a_6 = 7a_1

then α + β is equal to

(A)12
(B)3
(C)5
(D)7

Q26·MathematicsNumericalJEE Main 2026

Suppose a, b, c are in A.P. and

a^2

,

2b^2

,

c^2

are in G.P. If

a < b < c

and

a + b + c = 1

, then

9(a^2 + b^2 + c^2)

is equal to______ .

Q27·MathematicsSingle correctJEE Main 2026

If the sum of the first four terms of an A.P. is 6 and the sum of its first six terms is 4, then the sum of its first twelve terms is

(A)–20
(B)–24
(C)–26
(D)–22

Q28·MathematicsSingle correctJEE Main 2026

Let

a_1, \frac{a_2}{2}, \frac{a_3}{2^2}, \ldots, \frac{a_{10}}{2^9}

be a G.P. of common ratio

\frac{1}{\sqrt{2}}

. If

a_1 + a_2 + \ldots + a_{10} = 62

, then

a_1

is equal to :

(A) $2\left(\sqrt{2} - 1\right)$
(B) $2 - \sqrt{2}$
(C) $\sqrt{2} - 1$
(D) $2(2 - \sqrt{2})$

Q29·MathematicsNumericalJEE Main 2026

Let

a_{1}=1

and for

n\geq 1

,

a_{n+1}=\frac{1}{2}a_{n}+\frac{n^{2}-2n-1}{n^{2}(n+1)^{2}}

. Then

\left|\sum_{n=1}^{\infty}\left(a_{n}-\frac{2}{n^{2}}\right)\right|

is equal to ________.

Q30·MathematicsSingle correctJEE Main 2026

Let

a_{1}

,

a_{2}

,

a_{3}

,….. be a G.P. of increasing positive terms such that

a_{2}.a_{3}.a_{4} = 64

and

a_{1} + a_{3} + a_{5} = \frac{813}{7}

. Then

a_{3} + a_{5} + a_{7}

is equal to :

(A)3256
(B)3252
(C)3244
(D)3248

Q31·MathematicsNumericalJEE Advanced 2025

Let R denote the set of all real numbers. Let

f: R \to R

be a function such that

f(x) > 0

for all

x \in R

, and

f(x + y) = f(x) f(y)

for all

x, y \in R

. Let the real numbers

a_1

,

a_2

, .....,

a_{50}

be in an arithmetic progression. If

f(a_{31}) = 64 f(a_{25})

, and

\sum_{i=1}^{50} f(a_i) = 3\left(2^{25} + 1\right)

, then the value of

\sum_{i=6}^{30} f(a_i)

is ________ .

Q32·MathematicsSingle correctJEE Main 2025

If

\dfrac{1}{1^4}+\dfrac{1}{2^4}+\dfrac{1}{3^4}+\ldots\infty=\dfrac{\pi^4}{90}

,

\dfrac{1}{1^4}+\dfrac{1}{3^4}+\dfrac{1}{5^4}+\ldots\infty=\alpha

,

\dfrac{1}{2^4}+\dfrac{1}{4^4}+\dfrac{1}{6^4}+\ldots\infty=\beta

, then

\dfrac{\alpha}{\beta}

is equal to:

(A) $23$
(B) $18$
(C) $15$
(D) $14$

Q33·MathematicsSingle correctJEE Main 2025

Let

a_n

be the

n^{th}

term of an A.P. If

S_n=a_1+a_2+a_3+\dots+a_n=700

,

a_6=7

and

S_7=7

, then

a_n

is equal to:

(A)56
(B)65
(C)64
(D)70

Q34·MathematicsSingle correctJEE Main 2025

If the sum of the second, fourth and sixth terms of a G.P. of positive terms is 21 and the sum of its eighth, tenth and twelfth terms is 15309, then the sum of its first nine terms is:

(A)760
(B)755
(C)750
(D)757

Q35·MathematicsSingle correctJEE Main 2025

Let

x_1,x_2,x_3,x_4

be in a geometric progression. If 2, 7, 9, 5 are subtracted respectively from

x_1,x_2,x_3,x_4

then the resulting numbers are in an arithmetic progression. Then the value of

\dfrac{1}{24}(x_1x_2x_3x_4)

is:

(A)72
(B)18
(C)36
(D)216

Q36·MathematicsSingle correctJEE Main 2025

Consider two sets

A

and

B

, each containing three numbers in A.P. Let the sum and the product of the elements of

A

be 36 and

p

respectively and the sum and the product of the elements of

B

be 36 and

q

respectively. Let

d

and

D

be the common differences of the APs in

A

and

B

respectively such that

D=d+3

,

d>0

. If

\dfrac{p+q}{p-q}=\dfrac{19}{5}

, then

p-q

is equal to:

(A)600
(B)450
(C)630
(D)540

Q37·MathematicsSingle correctJEE Main 2025

Let

A=\{1,6,11,16,\ldots\}

and

B=\{9,16,23,30,\ldots\}

be the sets consisting of the first 2025 terms of two arithmetic progressions. Then

n(A\cup B)

is

(A)3814
(B)4027
(C)3761
(D)4003

Q38·MathematicsSingle correctJEE Main 2025

1+3+5^2+7+9^2+\ldots

upto 40 terms is equal to

(A)43890
(B)41880
(C)33980
(D)40870

Q39·MathematicsSingle correctJEE Main 2025

If the sum of the first 20 terms of the series

\dfrac{4\cdot1}{4+3\cdot1^2+1^4}+\dfrac{4\cdot2}{4+3\cdot2^2+2^4}+\dfrac{4\cdot3}{4+3\cdot3^2+3^4}+\dots

is

\dfrac{m}{n}

, where

m

and

n

are coprime, then

m+n

is equal to:

(A)423
(B)420
(C)421
(D)422

Q40·MathematicsSingle correctJEE Main 2025

The sum

1+3+11+25+45+71+\ldots

upto 20 terms, is equal to:

(A)7240
(B)7130
(C)6982
(D)8124

Q41·MathematicsSingle correctJEE Main 2025

The sum

1+\dfrac{1+3}{2!}+\dfrac{1+3+5}{3!}+\dfrac{1+3+5+7}{4!}+\ldots

upto

\infty

terms, is equal to:

(A) $6e$
(B) $4e$
(C) $3e$
(D) $2e$

Q42·MathematicsSingle correctJEE Main 2025

Let

a_1,a_2,a_3,\ldots

be a G.P. of increasing positive numbers. If

a_3a_5=729

and

a_2+a_4=\dfrac{111}{4}

, then

24(a_1+a_2+a_3)

is equal to:

(A)131
(B)130
(C)129
(D)128

Q43·MathematicsIntegerJEE Main 2025

If the sum of the first 10 terms of the series

\dfrac{4\cdot1}{1+4\cdot1^4}+\dfrac{4\cdot2}{1+4\cdot2^4}+\dfrac{4\cdot3}{1+4\cdot3^4}+\dots

is

\dfrac{m}{n}

, where

\gcd(m,n)=1

, then

m+n

is equal to ______.

Q44·MathematicsSingle correctJEE Main 2025

Let

a_1, a_2, a_3,\ldots

be in an A.P. such that

\displaystyle\sum_{k=1}^{12}a_{2k-1}=-\dfrac{72}{5}a_1

,

a_1\ne0

. If

\displaystyle\sum_{k=1}^{n}a_k=0

, then

n

is:

(A)11
(B)10
(C)18
(D)17

Q45·MathematicsSingle correctJEE Main 2025

The number of terms of an A.P. is even; the sum of all the odd terms is 24, the sum of all the even terms is 30 and the last term exceeds the first by

\dfrac{21}{2}

. Then the number of terms which are integers in the A.P. is:

(A)4
(B)10
(C)6
(D)8

Q46·MathematicsIntegerJEE Main 2025

Let

a_1, a_2,\ldots,a_{2024}

be an Arithmetic Progression such that

a_1+(a_5+a_{10}+a_{15}+\cdots+a_{2020})+a_{2024}=2233

. Then

a_1+a_2+a_3+\cdots+a_{2024}

is equal to ______.

Q47·MathematicsSingle correctJEE Main 2025

Consider an A.P. of positive integers, whose sum of the first three terms is 54 and the sum of the first twenty terms lies between 1600 and 1800. Then its

11^{\text{th}}

term is:

(A)84
(B)122
(C)90
(D)108

Q48·MathematicsSingle correctJEE Main 2025

The value of

\displaystyle\lim_{n\to\infty}\left(\sum_{k=1}^{n}\dfrac{k^3+6k^2+11k+5}{(k+3)!}\right)

is:

(A) $\dfrac{4}{3}$
(B) $2$
(C) $\dfrac{7}{3}$
(D) $\dfrac{5}{3}$

Q49·MathematicsSingle correctJEE Main 2025

Let

T_r

be the

r^{th}

term of an A.P. If for some m,

T_m=\frac{1}{25}

,

T_{25}=\frac{1}{20}

and

20\sum_{r=1}^{25} T_r=13

, then

5m\sum_{r=m}^{2m} T_r

is equal to:

(A)112
(B)126
(C)98
(D)142

Q50·MathematicsSingle correctJEE Main 2025

Let

\langle a_n\rangle

be a sequence such that

a_0=0

,

a_1=\frac{1}{2}

and

2a_{n+2}=5a_{n+1}-3a_n

,

n=0, 1, 2, 3, \ldots

Then

\sum_{k=1}^{100} a_k

is equal to:

(A) $3a_{99}-100$
(B) $3a_{100}-100$
(C) $3a_{100}+100$
(D) $3a_{99}+100$

Q51·MathematicsSingle correctJEE Main 2025

For positive integers n, if

4a_n=(n^2+5n+6)

and

S_n=\sum_{k=1}^{n}\frac{1}{a_k}

, then the value of

507\,S_{2025}

is:

(A)540
(B)1350
(C)675
(D)135

Q52·MathematicsIntegerJEE Main 2025

The interior angles of a polygon with n sides, are in an A.P. with common difference

6^\circ

. If the largest interior angle of the polygon is

219^\circ

, then n is equal to ______.

Q53·MathematicsSingle correctJEE Main 2025

In an arithmetic progression, if

S_{40}=1030

and

S_{12}=57

, then

S_{30}-S_{10}

is equal to:

(A)510
(B)515
(C)525
(D)505

Q54·MathematicsSingle correctJEE Main 2025

Let

S_n=\dfrac{1}{2}+\dfrac{1}{6}+\dfrac{1}{12}+\dfrac{1}{20}+\ldots

upto n terms. If the sum of the first six terms of an A.P. with first term

-p

and common difference

p

is

\sqrt{2026\,S_{2025}}

, then the absolute difference between

20^{th}

and

15^{th}

terms of the A.P. is

(A) $25$
(B) $90$
(C) $20$
(D) $45$

Q55·MathematicsSingle correctJEE Main 2025

If

7=5+\dfrac{1}{7}(5+\alpha)+\dfrac{1}{7^2}(5+2\alpha)+\dfrac{1}{7^3}(5+3\alpha)+\ldots\infty

, then the value of

\alpha

is:

(A)1
(B) $\dfrac{6}{7}$
(C)6
(D) $\dfrac{1}{7}$

Q56·MathematicsSingle correctJEE Main 2025

If the first term of an A.P. is 3 and the sum of its first four terms is equal to one-fifth of the sum of the next four terms, then the sum of the first 20 terms is equal to

(A) $-1200$
(B) $-1080$
(C) $-1020$
(D) $-120$

Q57·MathematicsIntegerJEE Main 2025

The roots of the quadratic equation

3x^2-px+q=0

are

10^{\text{th}}

and

11^{\text{th}}

terms of an arithmetic progression with common difference

\dfrac{3}{2}

. If the sum of the first

11

terms of this arithmetic progression is

88

, then

q-2p

is equal to __________.

Q58·MathematicsSingle correctJEE Main 2025

Let

a_1,a_2,\dots

be a G.P. of increasing positive terms. If

a_1a_5=28

and

a_2+a_4=29

, then

a_6

equals:

(A)628
(B)526
(C)784
(D)812

Q59·MathematicsIntegerJEE Main 2025

A finite sum of the form

\sum (r+2)/(2^{r+1})

-type series evaluates to

\tfrac{m}{n}

with

\gcd(m,n)=1

. Find

m-n

.

Q60·MathematicsSingle correctJEE Main 2025

Suppose that the number of terms in an A.P. is

2k

,

k\in\mathbb{N}

. If the sum of all odd terms of the A.P. is 40, the sum of all even terms is 55 and the last term of the A.P. exceeds the first term by 27, then

k

is equal to

(A)5
(B)8
(C)6
(D)4

Q61·MathematicsSingle correctJEE Main 2025

If

\displaystyle\sum_{r=1}^{n}T_r=\dfrac{(2n-1)(2n+1)(2n+3)(2n+5)}{64}

, then

\displaystyle\lim_{n\to\infty}\sum_{r=1}^{n}\dfrac1{T_r}

equals:

(A)1
(B)0
(C) $\tfrac23$
(D) $\tfrac13$

Q62·MathematicsNumericalJEE Main 2024

If a function

f

satisfies

f(m + n) = f(m) + f(n)

for all

m, n \in \mathbb{N}

and

f(1) = 1

, then the largest natural number

\lambda

such that

\sum_{k=1}^{2022} f(\lambda + k) \le (2022)^2

is equal to ________.

Q63·MathematicsSingle correctJEE Main 2024

Let

a

,

ar

,

ar^{2}

, ... be an infinite G.P. If

\displaystyle\sum_{n=0}^{\infty}ar^{n}=57

and

\displaystyle\sum_{n=0}^{\infty}a^{3}r^{3n}=9747

, then

a+18r

is equal to:

(A) $27$
(B) $46$
(C) $38$
(D) $31$

Q64·MathematicsSingle correctJEE Main 2024

If the sum of series

\frac{1}{1\cdot(1 + d)} + \frac{1}{(1 + d)(1 + 2d)} + \ldots + \frac{1}{(1 + 9d)(1 + 10d)}

is equal to 5, then

50d

is equal to:

(A)20
(B)5
(C)15
(D)10

Q65·MathematicsNumericalJEE Main 2024

If

\left(\dfrac{1}{\alpha+1}+\dfrac{1}{\alpha+2}+\cdots+\dfrac{1}{\alpha+1012}\right)-\left(\dfrac{1}{2\cdot 1}+\dfrac{1}{4\cdot 3}+\dfrac{1}{6\cdot 5}+\cdots+\dfrac{1}{2024\cdot 2023}\right)=\dfrac{1}{2024}

, then

\alpha

is equal to _______.

Q66·MathematicsNumericalJEE Main 2024

If the range of

f(\theta)=\dfrac{\sin^4\theta+3\cos^2\theta}{\sin^4\theta+\cos^2\theta}

,

\theta\in\mathbb{R}

is

[\alpha,\beta]

, then the sum of the infinite G.P., whose first term is

64

and the common ratio is

\dfrac\alpha\beta

, is equal to ___

Q67·MathematicsSingle correctJEE Main 2024

In an increasing geometric progression of positive terms, the sum of the second and sixth terms is

\dfrac{70}{3}

and the product of the third and fifth terms is

49

. Then the sum of the

4

th,

6

th and

8

th terms is

(A) $96$
(B) $78$
(C) $91$
(D) $84$

Q68·MathematicsNumericalJEE Main 2024

An arithmetic progression is written in the following way: row 1 is

2

; row 2 is

5,\ 8

; row 3 is

11,\ 14,\ 17

; row 4 is

20,\ 23,\ 26,\ 29

; and so on. The sum of all the terms of the

10

th row is _______ .

Q69·MathematicsNumericalJEE Main 2024

Let the positive integers be written in the form:

1

(row 1);

2,3

(row 2);

4,5,6

(row 3);

7,8,9,10

(row 4); and so on. If the

k^{th}

row contains exactly

k

numbers for every natural number

k

, then the row in which the number

5310

will be, is ___

Q70·MathematicsNumericalJEE Main 2024

Let the first term of a series be

T_{1}=6

and its

r^{\text{th}}

term

T_{r}=3T_{r-1}+6^{r}

,

r=2,3,\ldots,n

. If the sum of the first

n

terms of this series is

\dfrac{1}{5}(n^{2}-12n+39)(4\cdot 6^{n}-5\cdot 3^{n}+1)

, then

n

is equal to _______.

Q71·MathematicsSingle correctJEE Main 2024

A software company sets up m number of computer systems to finish an assignment in 17 days. If 4 computer systems crashed on the start of the second day, 4 more computer systems crashed on the start of the third day and so on, then it took 8 more days to finish the assignment. The value of m is equal to:

(A)125
(B)150
(C)180
(D)160

Q72·MathematicsSingle correctJEE Main 2024

Let ABC be an equilateral triangle. A new triangle is formed by joining the middle points of all sides of the triangle ABC and the same process is repeated infinitely many times. If P is the sum of perimeters and Q is be the sum of areas of all the triangles formed in this process, then:

(A) $P^2 = 36\sqrt{3}\,Q$
(B) $P^2 = 6\sqrt{3}\,Q$
(C) $P = 36\sqrt{3}\,Q^2$
(D) $P^2 = 72\sqrt{3}\,Q$

Q73·MathematicsNumericalJEE Main 2024

If

1+\dfrac{\sqrt{3}-\sqrt{2}}{2\sqrt{3}}+\dfrac{5-2\sqrt{6}}{18}+\dfrac{9\sqrt{3}-11\sqrt{2}}{36\sqrt{3}}+\dfrac{49-20\sqrt{6}}{180}+\cdots

upto

\infty=2\left(\sqrt{\dfrac{b}{a}}+1\right)\log_e\left(\dfrac{a}{b}\right)

, where

a

and

b

are integers with

\gcd(a,b)=1

, then

11a+18b

is equal to __________.

Q74·MathematicsSingle correctJEE Main 2024

If

\dfrac{1}{\sqrt{1}+\sqrt{2}} + \dfrac{1}{\sqrt{2}+\sqrt{3}} + \ldots + \dfrac{1}{\sqrt{99}+\sqrt{100}} = m

and

\dfrac{1}{1\cdot 2} + \dfrac{1}{2\cdot 3} + \ldots + \dfrac{1}{99\cdot 100} = n

, then the point

(m, n)

lies on the line:

(A) $11(x - 1) - 100(y - 2) = 0$
(B) $11(x - 2) - 100(y - 1) = 0$
(C) $11(x - 1) - 100y = 0$
(D) $11x - 100y = 0$

Q75·MathematicsSingle correctJEE Main 2024

For

x\ge 0

, the least value of

K

for which

4^{1+x}+4^{1-x}

,

\dfrac{K}{2}

,

16^x+16^{-x}

are three consecutive terms of an A.P. is equal to:

(A)10
(B)4
(C)8
(D)16

Q76·MathematicsNumericalJEE Main 2024

Let

a_1, a_2, a_3, \ldots

be in an arithmetic progression of positive terms. Let

A_k = a_1^2 - a_2^2 + a_3^2 - a_4^2 + \ldots + a_{2k-1}^2 - a_{2k}^2

. If

A_3 = -153

,

A_5 = -435

and

a_1^2 + a_2^2 + a_3^2 = 66

, then

a_{17} - A_7

is equal to ___.

Q77·MathematicsSingle correctJEE Main 2024

Let the first three terms

2

,

p

and

q

, with

q\ne2

, of a G.P. be respectively the

7^{th}

,

8^{th}

and

13^{th}

terms of an A.P. If the

5^{th}

term of the G.P. is the

n^{th}

term of the A.P., then

n

is equal to:

(A) $151$
(B) $169$
(C) $177$
(D) $163$

Q78·MathematicsSingle correctJEE Main 2024

The value of

\dfrac{1\cdot 2^2+2\cdot 3^2+\ldots+100\cdot(101)^2}{1^2\cdot 2+2^2\cdot 3+\ldots+100^2\cdot 101}

is

(A) $\tfrac{306}{305}$
(B) $\tfrac{305}{301}$
(C) $\tfrac{32}{31}$
(D) $\tfrac{31}{30}$

Q79·MathematicsSingle correctJEE Main 2024

Let three real numbers

a

,

b

,

c

be in arithmetic progression and

a+1

,

b

,

c+3

be in geometric progression. If

a>10

and the arithmetic mean of

a

,

b

and

c

is 8, then the cube of the geometric mean of

a

,

b

and

c

is

(A)120
(B)312
(C)316
(D)128

Q80·MathematicsNumericalJEE Main 2024

Let

a=1+\dfrac{{}^2C_2}{3!}+\dfrac{{}^3C_2}{4!}+\dfrac{{}^4C_2}{5!}+\ldots

and

b=1+\dfrac{{}^1C_0+{}^1C_1+{}^2C_2}{1!}+\dfrac{{}^2C_0+{}^3C_1+{}^4C_2}{2!}+\dfrac{{}^3C_0+{}^4C_1+{}^5C_2}{3!}+\ldots

. Then

\dfrac{2b}{a^2}

is equal to ___

Q81·MathematicsSingle correctJEE Main 2024

Let

S_n

denote the sum of the first

n

terms of an arithmetic progression. If

S_{10}=390

and the ratio of the tenth and the fifth terms is

15:7

, then

S_{15}-S_5

is equal to:

(A)800
(B)890
(C)790
(D)690

Q82·MathematicsNumericalJEE Main 2024

Let

3,7,11,15,\ldots,403

and

2,5,8,11,\ldots,404

be two arithmetic progressions. Then the sum, of the common terms in them, is equal to ___

Q83·MathematicsSingle correctJEE Main 2024

Let

3,a,b,c

be in A.P. and

3,a-1,b+1,c+9

be in G.P. Then, the arithmetic mean of

a,b,c

is:

(A) $-4$
(B) $-1$
(C) $13$
(D) $11$

Q84·MathematicsNumericalJEE Main 2024

If three successive terms of a G.P. with common ratio

r

(r>1)

are the lengths of the sides of a triangle and

[r]

denotes the greatest integer less than or equal to

r

, then

3[r]+[-r]

is equal to __________.

Q85·MathematicsSingle correctJEE Main 2024

The sum of the series

\dfrac{1}{1-3\cdot 1^2+1^4}+\dfrac{2}{1-3\cdot 2^2+2^4}+\dfrac{3}{1-3\cdot 3^2+3^4}+\cdots

up to 10 terms is

(A) $\dfrac{45}{109}$
(B) $-\dfrac{45}{109}$
(C) $\dfrac{55}{109}$
(D) $-\dfrac{55}{109}$

Q86·MathematicsSingle correctJEE Main 2024

Let

2^{\text{nd}}

,

8^{\text{th}}

and

44^{\text{th}}

terms of a non-constant A.P. be respectively the

1^{\text{st}}

,

2^{\text{nd}}

and

3^{\text{rd}}

terms of G.P. If the first term of A.P. is 1, then the sum of first 20 terms is equal to

(A)980
(B)960
(C)990
(D)970

Q87·MathematicsNumericalJEE Main 2024

Let

S_n

be the sum to

n

-terms of an arithmetic progression

3,7,11,\ldots

. If

40<\dfrac{6}{n(n+1)}\displaystyle\sum_{k=1}^{n}S_k<42

, then

n

equals ______.

Q88·MathematicsSingle correctJEE Main 2024

Let

S_n

denote the sum of first

n

terms of an arithmetic progression. If

S_{20}=790

and

S_{10}=145

, then

S_{15}-S_5

is:

(A) $395$
(B) $390$
(C) $405$
(D) $410$

Q89·MathematicsSingle correctJEE Main 2024

Let

a

and

b

be two distinct positive real numbers. Let

11^{\text{th}}

term of a GP, whose first term is

a

and third term is

b

, is equal to

p^{\text{th}}

term of another GP, whose first term is

a

and fifth term is

b

. Then

p

is equal to:

(A) $20$
(B) $25$
(C) $21$
(D) $24$

Q90·MathematicsNumericalJEE Main 2024

Let

\alpha=1^2+4^2+8^2+13^2+19^2+26^2+\ldots

upto

10

terms and

\beta=\displaystyle\sum_{n=1}^{10}n^4

. If

4\alpha-\beta=55k+40

, then

k

is equal to ___

Q91·MathematicsSingle correctJEE Main 2024

If in a G.P. of 64 terms, the sum of all the terms is 7 times the sum of the odd terms of the G.P., then the common ratio of the G.P. is equal to

(A)7
(B)4
(C)5
(D)6

Q92·MathematicsSingle correctJEE Main 2024

If each term of a geometric progression

a_1,a_2,a_3,\ldots

with

a_1=\dfrac{1}{8}

and

a_2\neq a_1

, is the arithmetic mean of the next two terms and

S_n=a_1+a_2+\ldots+a_n

, then

S_{20}-S_{18}

is equal to:

(A) $2^{15}$
(B) $-2^{18}$
(C) $2^{18}$
(D) $-2^{15}$

Q93·MathematicsSingle correctJEE Main 2024

If

\log_e a

,

\log_e b

,

\log_e c

are in an A.P. and

\log_e a-\log_e 2b

,

\log_e 2b-\log_e 3c

,

\log_e 3c-\log_e a

are also in an A.P., then

a:b:c

is equal to:

(A) $9:6:4$
(B) $16:4:1$
(C) $25:10:4$
(D) $6:3:2$

Q94·MathematicsSingle correctJEE Main 2024

In an A.P., the sixth term

a_6=2

. If the

a_1 a_4 a_5

is the greatest, then the common difference of the A.P., is equal to

(A) $\dfrac{3}{2}$
(B) $\dfrac{8}{5}$
(C) $\dfrac{2}{3}$
(D) $\dfrac{5}{8}$

Q95·MathematicsSingle correctJEE Main 2024

If

\alpha,\beta

are the roots of the equation

x^2-x-1=0

and

S_n=2023\alpha^n+2024\beta^n

, then

(A) $2S_{12}=S_{11}+S_{10}$
(B) $S_{12}=S_{11}+S_{10}$
(C) $2S_{11}=S_{12}+S_{10}$
(D) $S_{11}=S_{10}+S_{12}$

Q96·MathematicsSingle correctJEE Main 2024

The number of common terms in the progressions

4,9,14,19,\ldots

, up to

25^{\text{th}}

term and

3,6,9,12,\ldots

, up to

37^{\text{th}}

term is:

(A)9
(B)5
(C)7
(D)8

Q97·MathematicsNumericalJEE Main 2024

If

8=3+\dfrac{1}{4}(3+p)+\dfrac{1}{4^2}(3+2p)+\dfrac{1}{4^3}(3+3p)+\ldots\infty

, then the value of

p

is __________.

Q98·MathematicsSingle correctJEE Main 2024

The

20^{\text{th}}

term from the end of the progression

20,19\dfrac14,18\dfrac12,17\dfrac34,\ldots,-129\dfrac14

is :

(A) $-118$
(B) $-110$
(C) $-115$
(D) $-100$

Q99·MathematicsNumericalJEE Advanced 2023

Let

7\overbrace{5\cdots5}^{r}7

denote the (r + 2) digit number where the first and the last digits are 7 and the remaining r digits are 5. Consider the sum

S = 77 + 757 + 7557 + \ldots + 7\overbrace{5\cdots5}^{98}7

. If

S = \frac{7\overbrace{5\cdots5}^{99}7 + m}{n}

, where m and n are natural numbers less than 3000, then the value of m + n is

Q100·MathematicsNumericalJEE Main 2023

If the sum of the series

\left(\frac{1}{2} - \frac{1}{3}\right) + \left(\frac{1}{2^2} - \frac{1}{2 \cdot 3} + \frac{1}{3^2}\right) + \left(\frac{1}{2^3} - \frac{1}{2^2 \cdot 3} + \frac{1}{2 \cdot 3^2} - \frac{1}{3^3}\right) + \left(\frac{1}{2^4} - \frac{1}{2^3 \cdot 3} + \frac{1}{2^2 \cdot 3^2} - \frac{1}{2 \cdot 3^3} + \frac{1}{3^4}\right) + \cdots

is

\frac{\alpha}{\beta}

, where

\alpha

and

\beta

are co-prime, then

\alpha + 3\beta

is equal to....

Q101·MathematicsSingle correctJEE Main 2023

Let

A_1

and

A_2

be two arithmetic means and

G_1, G_2, G_3

be three geometric means of two distinct positive numbers. The

G_1^4+G_2^4+G_3^4+G_1^2 G_3^2

is equal to

(A) $2(A_1+A_2)G_1 G_3$
(B) $(A_1+A_2)^2 G_1 G_3$
(C) $(A_1+A_2)G_1^2 G_3^2$
(D) $2(A_1+A_2)G_1^2 G_3^2$

Q102·MathematicsSingle correctJEE Main 2023

Let

a_1, a_2, a_3, \ldots

be a G.P. of increasing positive numbers. Let the sum of its

6^{th}

and

8^{th}

terms be 2 and the product of its

3^{rd}

and

5^{th}

terms be

\dfrac{1}{9}

. Then

6(a_2 + a_4)(a_4 + a_6)

is equal to

(A) $2\sqrt{2}$
(B) $2$
(C) $3\sqrt{3}$
(D) $3$

Q103·MathematicsNumericalJEE Main 2023

Let

[\alpha]

denote the greatest integer

\le \alpha

. Then

\left[\sqrt{1}\right] + \left[\sqrt{2}\right] + \left[\sqrt{3}\right] + \ldots + \left[\sqrt{120}\right]

is equal to

Q104·MathematicsSingle correctJEE Main 2023

Among: (S1):

\displaystyle\lim_{n\to\infty}\dfrac{1}{n^{2}}(2+4+6+\dots+2n)=1

(S2):

\displaystyle\lim_{n\to\infty}\dfrac{1}{n^{16}}(1^{15}+2^{15}+3^{15}+\dots+n^{15})=\dfrac{1}{16}

(A)Both (S1) and (S2) are false
(B)Both (S1) and (S2) are true
(C)Only (S2) is true
(D)Only (S1) is true

Q105·MathematicsNumericalJEE Main 2023

The sum to

20

terms of the series

2\cdot 2^{2}-3^{2}+2\cdot 4^{2}-5^{2}+2\cdot 6^{2}-\dots

is equal to _____.

Q106·MathematicsNumericalJEE Main 2023

Let

f(x) = \sum\limits_{k=1}^{10} k x^k

,

x \in \mathbb{R}

. If

2f(2) + f'(2) = 119(2)^k + 1

then

k

is equal to

Q107·MathematicsSingle correctJEE Main 2023

Let

s_{1},s_{2},s_{3},\dots,s_{10}

respectively be the sum to

12

terms of

10

A.P.'s whose first terms are

1,2,3,\dots,10

and the common differences are

1,3,5,\dots,19

respectively. Then

\displaystyle\sum_{i=1}^{10}s_{i}

is equal to:

(A) $7380$
(B) $7220$
(C) $7360$
(D) $7260$

Q108·MathematicsNumericalJEE Main 2023

Let the positive numbers

a_1,a_2,a_3,a_4

and

a_5

be in a G.P. Let their mean and variance be

\dfrac{31}{10}

and

\dfrac{m}{n}

respectively, where m and n are co-prime. If the mean of their reciprocals is

\dfrac{31}{40}

and

a_3+a_4+a_5=14

, then

m+n

is equal to _________.

Q109·MathematicsSingle correctJEE Main 2023

Let

\langle a_n\rangle

be a sequence such that

a_1+a_2+\cdots+a_n=\dfrac{n^2+3n}{(n+1)(n+2)}

. If

28\sum_{k=1}^{10}\dfrac{1}{a_k}=p_1 p_2 p_3\cdots p_m

, where

p_1,p_2,\ldots,p_m

are the first m prime numbers, then m is equal to

(A)7
(B)6
(C)5
(D)8

Q110·MathematicsNumericalJEE Main 2023

For

k\in\mathbb{N}

, if the sum of the series

1+\dfrac{4}{k}+\dfrac{8}{k^2}+\dfrac{13}{k^3}+\dfrac{19}{k^4}+\cdots

is

10

, then the value of

k

is

Q111·MathematicsSingle correctJEE Main 2023

Let

x_1, x_2, \ldots, x_{100}

be in an arithmetic progression, with

x_1 = 2

and their mean equal to

200

. If

y_i = i(x_i - i)

,

1 \le i \le 100

, then the mean of

y_1, y_2, \ldots, y_{100}

is

(A) $10101.50$
(B) $10051.50$
(C) $10049.50$
(D) $10100$

Q112·MathematicsNumericalJEE Main 2023

Let

S = 109 + \frac{108}{5} + \frac{107}{5^2} + \ldots + \frac{2}{5^{107}} + \frac{1}{5^{108}}

. Then the value of

(16S - (25)^{-54})

is equal to _______ .

Q113·MathematicsSingle correctJEE Main 2023

If

S_n=4+11+21+34+50+\ldots

to

n

terms, then

\frac{1}{60}\left(S_{29}-S_6\right)

is equal to:

(A) $226$
(B) $220$
(C) $223$
(D) $227$

Q114·MathematicsNumericalJEE Main 2023

Suppose

a_1,a_2,2,a_3,a_4

be in an arithmetico-geometric progression. If the common ratio of the corresponding geometric progression is

2

and the sum of all

5

terms of the arithmetico-geometric progression is

\frac{49}{2}

, then

a_4

is equal to _______ .

Q115·MathematicsNumericalJEE Main 2023

The sum of all those terms, of the arithmetic progression

3,8,13,\ldots,373

, which are not divisible by 3, is equal to _________.

Q116·MathematicsSingle correctJEE Main 2023

Let the first term

a

and the common ratio

r

of a geometric progression be positive integers. If the sum of squares of its first three terms is 33033, then the sum of these three terms is equal to

(A) $231$
(B) $210$
(C) $220$
(D) $241$

Q117·MathematicsSingle correctJEE Main 2023

Let

a_n

be the

n^{th}

term of the series

5+8+14+23+35+50+\ldots

and

S_n=\sum_{k=1}^{n}a_k

. Then

S_{30}-a_{40}

is equal to

(A)11310
(B)11280
(C)11290
(D)11260

Q118·MathematicsNumericalJEE Main 2023

If

a_{n}

is the greatest term in the sequence

a_{n}=\frac{n^{3}}{n^{4}+147}

,

n=1,2,3,\ldots

, then

\alpha

is equal to

Q119·MathematicsNumericalJEE Main 2023

Let

0<z<y<x

be three real numbers such that

\dfrac{1}{x},\dfrac{1}{y},\dfrac{1}{z}

are in an arithmetic progression and

x,\sqrt{2}y,z

are in a geometric progression. If

xy+yz+zx=\dfrac{3}{\sqrt{2}}xyz

, then

3(x+y+z)^{2}

is equal to

Q120·MathematicsSingle correctJEE Main 2023

Let

S_{k}=\frac{1+2+\dots+K}{K}

and

\sum_{j=1}^{n} S_{j}^{2}=\frac{n}{A}(Bn^{2}+Cn+D)

, where

A,B,C,D\in N

and

A

has least value. Then

(A) $A+B$ is divisible by $D$
(B) $A+B=5(D-C)$
(C) $A+C+D$ is not divisible by $B$
(D) $A+B+C+D$ is divisible by 5

Q121·MathematicsSingle correctJEE Main 2023

The sum of the first

20

terms of the series

5+11+19+29+41+\dots

is:

(A)3450
(B)3250
(C)3420
(D)3520

Q122·MathematicsNumericalJEE Main 2023

If

(20)^{19}+2(21)(20)^{18}+3(21)^2(20)^{17}+\cdots+20(21)^{19}=k(20)^{18}

then

k

is equal to

Q123·MathematicsSingle correctJEE Main 2023

Let

a_1,a_2,\dots,a_n

be

n

positive consecutive terms of an arithmetic progression. If

d>0

is its common difference, then

\displaystyle\lim_{n\to\infty}\sqrt{\dfrac{d}{n}}\left(\dfrac{1}{\sqrt{a_1}+\sqrt{a_2}}+\dfrac{1}{\sqrt{a_2}+\sqrt{a_3}}+\dots+\dfrac{1}{\sqrt{a_{n-1}}+\sqrt{a_n}}\right)

is:

(A) $\sqrt d$
(B) $0$
(C) $\dfrac{1}{\sqrt d}$
(D) $\infty$

Q124·MathematicsSingle correctJEE Main 2023

If

\gcd(m,n)=1

and

1^2-2^2+3^2-4^2+\cdots+(2021)^2-(2022)^2+(2023)^2=1012\,m^2n

then

m^2-n^2

is equal to

(A)240
(B)220
(C)210
(D)180

Q125·MathematicsNumericalJEE Main 2023

The sum of the common terms of the following three arithmetic progressions:

3,7,11,15,\ldots,399

;

2,5,8,11,\ldots,359

and

2,7,12,17,\ldots,197

is equal to

Q126·MathematicsNumericalJEE Main 2023

Let

a_1, a_2, a_3, \ldots, a_n

be in A.P. If the sum of its first four terms is

50

and the sum of its last four terms is

170

, then the product of its middle two terms is _______ .

Q127·MathematicsSingle correctJEE Main 2023

The sum to

10

terms of the series

\frac{1}{1+1^2+1^4}+\frac{2}{1+2^2+2^4}+\frac{3}{1+3^2+3^4}+\cdots

is

(A) $\frac{55}{111}$
(B) $\frac{56}{111}$
(C) $\frac{58}{111}$
(D) $\frac{59}{111}$

Q128·MathematicsSingle correctJEE Main 2023

The sum

\sum_{n=1}^{\infty}\dfrac{2n^2+3n+4}{(2n)!}

is equal to:

(A) $\dfrac{13e}{4}+\dfrac{5}{4e}$
(B) $\dfrac{11e}{2}+\dfrac{7}{2e}-4$
(C) $\dfrac{11e}{2}+\dfrac{7}{2e}$
(D) $\dfrac{13e}{4}+\dfrac{5}{4e}-4$

Q129·MathematicsNumericalJEE Main 2023

Let

a_1, a_2, \ldots, a_n

be in A.P. If

a_5 = 2a_7

and

a_{11} = 18

, then

12\left(\frac{1}{\sqrt{a_{10}} + \sqrt{a_{11}}} + \frac{1}{\sqrt{a_{11}} + \sqrt{a_{12}}} + \cdots + \frac{1}{\sqrt{a_{17}} + \sqrt{a_{18}}}\right)

is equal to

Q130·MathematicsSingle correctJEE Main 2023

Let

a_1,a_2,a_3,\ldots

be an A.P. If

a_7=3

, the product

a_1a_4

is minimum and the sum of its first

n

terms is zero, then

n!-4a_{n(n+2)}

is equal to:

(A) $9$
(B) $\dfrac{33}{4}$
(C) $\dfrac{381}{4}$
(D) $24$

Q131·MathematicsNumericalJEE Main 2023

The sum

1^2-2\cdot3^2+3\cdot5^2-4\cdot7^2+5\cdot9^2-\cdots+15\cdot29^2

is

Q132·MathematicsSingle correctJEE Main 2023

If the sum and product of four positive consecutive terms of a G.P., are 126 and 1296, respectively, then the sum of common ratios of all such GPs is

(A) $7$
(B) $3$
(C) $\frac{9}{2}$
(D) $14$

Q133·MathematicsSingle correctJEE Main 2023

Let

\alpha \in (0,1)

and

\beta = \log_e(1 - \alpha)

. Let

P_n(x) = x + \frac{x^2}{2} + \frac{x^3}{3} + \cdots + \frac{x^n}{n},\ x \in (0,1)

. Then the integral

\int_0^{\alpha} \frac{t^{50}}{1 - t}\, dt

is equal to

(A) $\beta + P_{50}(\alpha)$
(B) $P_{50}(\alpha) - \beta$
(C) $\beta - P_{50}(\alpha)$
(D) $-(\beta + P_{50}(\alpha))$

Q134·MathematicsNumericalJEE Main 2023

Let

\sum_{n=0}^{\infty}\dfrac{n^3((2n)!)+(2n-1)(n!)}{(n!)((2n)!)}=ae+\dfrac{b}{e}+c

, where

a,b,c\in\mathbb{Z}

and

e=\sum_{n=0}^{\infty}\dfrac{1}{n!}

. Then

a^2-b+c

is equal to

Q135·MathematicsNumericalJEE Main 2023

The

8^{th}

common term of the series

S_1=3 + 7 + 11 + 15 + 19 + \cdots

and

S_2=1 + 6 + 11 + 16 + 21 + \cdots

is _______.

Q136·MathematicsSingle correctJEE Main 2023

Let

a, b, c > 1

;

a^3, b^3

and

c^3

be in A.P., and

\log_a b, \log_c a

and

\log_b c

be in G.P. If the sum of the first 20 terms of an A.P., whose first term is

\dfrac{a + 4b + c}{3}

and the common difference is

\dfrac{a - 8b + c}{10}

, is

-444

, then

abc

is equal to:

(A) $\dfrac{125}{8}$
(B) $216$
(C) $343$
(D) $\dfrac{343}{8}$

Q137·MathematicsSingle correctJEE Main 2023

If

a_n=\dfrac{-2}{4n^2-16n+15}

, then

a_1+a_2+\cdots+a_{25}

is equal to:

(A) $\dfrac{52}{147}$
(B) $\dfrac{49}{138}$
(C) $\dfrac{50}{141}$
(D) $\dfrac{51}{144}$

Q138·MathematicsSingle correctJEE Main 2023

Suppose

f:\mathbb{R}\to(0,\infty)

be a differentiable function such that

5f(x+y)=f(x)\cdot f(y)

,

\forall x,y\in\mathbb{R}

. If

f(3)=320

, then

\sum_{n=0}^{5}f(n)

is equal to:

(A) $6875$
(B) $6525$
(C) $6825$
(D) $6575$

Q139·MathematicsNumericalJEE Main 2023

Let

a_1, a_2, a_3, \ldots

be a GP of increasing positive numbers. If the product of fourth and sixth terms is 9 and the sum of fifth and seventh terms is 24, then

a_1 a_9 + a_2 a_4 a_9 + a_5 + a_7

is equal to ________ .

Q140·MathematicsNumericalJEE Main 2023

Let

a_1=b_1=1

and

a_n=a_{n-1}+(n-1),\,b_n=b_{n-1}+a_{n-1},\,\forall n\ge 2

. If

S=\sum_{n=1}^{10}\dfrac{b_n}{2^n}

and

T=\sum_{n=1}^{8}\dfrac{n}{2^{n-1}}

, then

2^{7}(2S-T)

is equal to _____.

Q141·MathematicsNumericalJEE Main 2023

Let

\{a_k\}

and

\{b_k\},\,k\in\mathbb{N}

, be two G.P.s with common ratios

r_1

and

r_2

respectively such that

a_1=b_1=4

and

r_1<r_2

. Let

c_k=a_k+b_k,\,k\in\mathbb{N}

. If

c_2=5

and

c_3=\dfrac{13}{4}

then

\sum_{k=1}^{\infty}c_k-(12a_6+8b_4)

is equal to _____.

Q142·MathematicsNumericalJEE Main 2023

Suppose

f

is a function satisfying

f(x + y) = f(x) + f(y)

for all

x, y \in \mathbb{N}

and

f(1) = \dfrac{1}{5}

. If

\sum_{n=1}^{m} \dfrac{f(n)}{n(n + 1)(n + 2)} = \dfrac{1}{12}

, then

m

is equal to ________ .

Q143·MathematicsSingle correctJEE Main 2023

Consider a function

f:\mathbb{N}\to\mathbb{R}

, satisfying

f(1)+2f(2)+3f(3)+\dots+xf(x)=x(x+1)f(x);\,x\ge 2

with

f(1)=1

. Then

\dfrac{1}{f(2022)}+\dfrac{1}{f(2028)}

is equal to:

(A) $8100$
(B) $8400$
(C) $8000$
(D) $8200$

Q144·MathematicsNumericalJEE Main 2023

Let

A_1,A_2,A_3

be three A.P. with the same common difference

d

and having their first terms as

A,A+1,A+2

, respectively. Let

a,b,c

be the

7^{th},9^{th},17^{th}

terms of

A_1,A_2,A_3

, respectively, such that

\begin{vmatrix}a & 7 & 1\\ 2b & 17 & 1\\ c & 17 & 1\end{vmatrix}+70=0

. If

a=29

, then the sum of the first

20

terms of an AP whose first term is

c-a-b

and common difference is

\dfrac{d}{12}

, is equal to _______.

Q145·MathematicsNumericalJEE Main 2023

For two positive numbers

a,b

such that

a,b

and

\dfrac{1}{18}

are in a geometric progression, while

\dfrac{1}{a},10

and

\dfrac{1}{b}

are in an arithmetic progression, then

16a+b

is equal to

Q146·MathematicsNumericalJEE Main 2023

If

\dfrac{1^3+2^3+3^3+\cdots\text{ up to }n\text{ terms}}{1\cdot3+2\cdot5+3\cdot7+\cdots\text{ up to }n\text{ terms}}=\dfrac{9}{5}

, then the value of

n

is

Q147·MathematicsNumericalJEE Main 2023

The

4^{\text{th}}

term of GP is

500

and its common ratio is

\dfrac{1}{m}

,

m\in\mathbb{N}

. Let

S_n

denote the sum of the first

n

terms of this GP. If

S_6>S_5+1

and

S_7<S_6+\tfrac{1}{2}

, then the number of possible values of

m

is _______ .

Q148·MathematicsSingle correctJEE Main 2023

For three positive integers

p,q,r

,

x^{pq^2}=y^{qr}=z^{p^2 r}

and

r=pq+1

such that

3,\ 3\log_y x,\ 3\log_z y,\ 7\log_x z

are in A.P. with common difference

\tfrac{1}{2}

. Then

r-p-q

is equal to

(A) $-6$
(B) $12$
(C) $6$
(D) $2$

Q149·MathematicsSingle correctJEE Main 2023

Let

f(x)

be a function such that

f(x+y)=f(x)\cdot f(y)

for all

x,y\in\mathbb{N}

. If

f(1)=3

and

\sum_{k=1}^{n}f(k)=3279

, then the value of

n

is

(A) $9$
(B) $6$
(C) $8$
(D) $7$

Q150·MathematicsMultiple correctJEE Advanced 2022

Let

a_{1}, a_{2}, a_{3}, .....

be an arithmetic progression with

a_{1} = 7

and common difference 8. Let

T_{1}, T_{2}, T_{3}, .....

be such that

T_{1} = 3

and

T_{n+1} - T_{n} = a_{n}

for

n \geq 1

. Then, which of the following is/are TRUE?

(A) $T_{20} = 1604$
(B) $\sum_{k=1}^{20} T_{k} = 10510$
(C) $T_{30} = 3454$
(D) $\sum_{k=1}^{30} T_{k} = 35610$

Q151·MathematicsNumericalJEE Advanced 2022

Let

l_{1}, l_{2}, ....., l_{100}

be consecutive terms of an arithmetic with common difference

d_{1}

, and let

w_{1}, w_{2}, ....., w_{100}

be consecutive terms of another arithmetic progression with common difference

d_{2}

, where

d_{1}d_{2} = 10

. For each

i

= 1, 2, ....., 100, let

R_{i}

be a rectangle with length

l_{i}

, width

w_{i}

and area

A_{i}

. If

A_{51} - A_{50} = 1000

, then the value of

A_{100} - A_{90}

is __________.

Q152·MathematicsSingle correctJEE Main 2022

If

\frac{1}{(20 - a)\,(40 - a)} + \frac{1}{(40 - a)\,(60 - a)} + \ldots\ldots+ \frac{1}{(180 - a)\,(200 - a)} = \frac{1}{256}

, then the maximum value of a is :

(A)198
(B)202
(C)212
(D)218

Q153·MathematicsSingle correctJEE Main 2022

Let

\{a_n\}_{n=0}^{\infty}

be a sequence such that

a_0 = a_1 = 0

and

a_{n+2} = 3a_{n+1} - 2a_n + 1

,

\forall\ n \geq 0

. Then

a_{25}\, a_{23} - 2\, a_{25}\, a_{22} - 2\, a_{23}\, a_{24} + 4\, a_{22}\, a_{24}

is equal to:

(A) $483$
(B) $528$
(C) $575$
(D) $624$

Q154·MathematicsNumericalJEE Main 2022

Let

a_{1},\,a_{2},\,a_{3},\ldots

be an A.P. If

\sum_{r=1}^{\infty}\frac{a_{r}}{2^{r}} = 4

, then

4a_{2}

is equal to __________.

Q155·MathematicsNumericalJEE Main 2022

If

\frac{1}{2 \times 3 \times 4} + \frac{1}{3 \times 4 \times 5} + \frac{1}{4 \times 5 \times 6} + \ldots + \frac{1}{100 \times 101 \times 102} = \frac{k}{101}

, then 34 k is equal to _____.

Q156·MathematicsSingle correctJEE Main 2022

\sum_{r=1}^{20} \left( r^2 + 1 \right)\left( r! \right)

is equal to:

(A) $22! - 21!$
(B) $22! - 2\,(21!)$
(C) $21! - 2\,(20!)$
(D) $21! - 20!$

Q157·MathematicsNumericalJEE Main 2022

If

\frac{6}{3^{12}}+\frac{10}{3^{11}}+\frac{20}{3^{10}}+\frac{40}{3^{9}}+\ldots+\frac{10240}{3}=2^{n}\cdot m

, where m is odd, then

m.n

is equal to _____

Q158·MathematicsSingle correctJEE Main 2022

Consider the sequence

a_{1}

,

a_{2}

,

a_{3}

, …… such that

a_{1} = 1

,

a_{2} = 2

and

a_{n+2} = \frac{2}{a_{n+1}} + a_{n}

for n = 1, 2, 3, … . If

\left( \frac{a_{1} + \frac{1}{a_{2}}}{a_{3}} \right) \cdot \left( \frac{a_{2} + \frac{1}{a_{3}}}{a_{4}} \right) \cdot \left( \frac{a_{3} + \frac{1}{a_{4}}}{a_{5}} \right) \cdots \left( \frac{a_{30} + \frac{1}{a_{31}}}{a_{32}} \right) = 2^{\alpha} \left( {}^{61}C_{31} \right)

, then

\alpha

is equal to :

(A)−30
(B)−31
(C)−60
(D)−61

Q159·MathematicsNumericalJEE Main 2022

\frac{2^{3}-1^{3}}{1 \times 7} + \frac{4^{3}-3^{3}+2^{3}-1^{3}}{2 \times 11} + \frac{6^{3}-5^{3}+4^{3}-3^{3}+2^{3}-1^{3}}{3 \times 15} + ..... + \frac{30^{3}-29^{3}+28^{3}-27^{3}+...+2^{3}-1^{3}}{15 \times 63}

is equal to _____.

Q160·MathematicsSingle correctJEE Main 2022

Let the sum of an infinite G.P., whose first term is a and the common ratio is r, be 5. Let the sum of its first five terms be

\frac{98}{25}

. Then the sum of the first 21 terms of an AP, whose first term is

10ar

,

n^{th}

term is

a_{n}

and the common difference is

10ar^{2}

, is equal to :

(A) $21\,a_{11}$
(B) $22\,a_{11}$
(C) $15\,a_{16}$
(D) $14\,a_{16}$

Q161·MathematicsSingle correctJEE Main 2022

Suppose

a_1, a_2

, ....,

a_n

,... be an arithmetic progression of natural numbers. If the ratio of the sum of the first five terms of the sum of first nine terms of the progression is 5 : 17 and

110 < a_{15} < 120

, then the sum of the first ten terms of the progression is equal to -

(A)290
(B)380
(C)460
(D)510

Q162·MathematicsSingle correctJEE Main 2022

Consider two G.Ps. 2, 2

^{2}

, 2

^{3}

, ... and 4, 4

^{2}

, 4

^{3}

,.... of 60 and n terms respectively. If the geometric mean of all the 60 + n terms is

(2)^{\frac{225}{8}}

, then

\sum\limits_{k=1}^{n} k(n-k)

is equal to :

(A)560
(B)1540
(C)1330
(D)2600

Q163·MathematicsNumericalJEE Main 2022

If

\sum_{k=1}^{10}\frac{k}{k^{4}+k^{2}+1}=\frac{m}{n}

, where m and n are co-prime, then

m+n

is equal to

Q164·MathematicsNumericalJEE Main 2022

The series of positive multiples of 3 is divided into sets : {3}, {6, 9,12}, {15, 18, 21, 24, 27},... Then the sum of the elements in the 11

^{th}

set is equal to__________,

Q165·MathematicsNumericalJEE Main 2022

Different A.P.'s are constructed with the first term 100, the last term 199, And integral common differences. The sum of the common differences of all such, A.P's having at least 3 terms and at most 33 terms is.

Q166·MathematicsNumericalJEE Main 2022

Let

a_{1} = b_{1} = 1

,

a_{n} = a_{n-1} + 2

and

b_{n} = a_{n} + b_{n-1}

for every natural number

n \ge 2

. Then

\sum_{n=1}^{15} a_{n} \cdot b_{n}

is equal to ________.

Q167·MathematicsSingle correctJEE Main 2022

The sum of the infinite series

1 + \frac{5}{6} + \frac{12}{6^{2}} + \frac{22}{6^{3}} + \frac{35}{6^{4}} + \frac{51}{6^{5}} + \frac{70}{6^{6}} + ....

is equal to:

(A) $\frac{425}{216}$
(B) $\frac{429}{216}$
(C) $\frac{288}{125}$
(D) $\frac{280}{125}$

Q168·MathematicsNumericalJEE Main 2022

Let 3, 6, 9, 12,... upto 78 terms and 5, 9, 13, 17,... upto 59 terms be two series. Then, the sum of the terms common to both the series is equal to____.

Q169·MathematicsSingle correctJEE Main 2022

Let

\{a_{n}\}_{n=0}^{\infty}

be a sequence such that

a_{0} = a_{1} = 0

and

a_{n+2} = 2a_{n+1} - a_{n} + 1

for all

n \ge 0

. Then,

\sum_{n=2}^{\infty}\frac{a_{n}}{7^{n}}

is equal to

(A) $\frac{6}{343}$
(B) $\frac{7}{216}$
(C) $\frac{8}{343}$
(D) $\frac{49}{216}$

Q170·MathematicsSingle correctJEE Main 2022

Let

\mathrm{A}_{1},

\mathrm{A}_{2},

\mathrm{A}_{3},

…… be an increasing geometric progression of positive real numbers. If

\mathrm{A}_{1}\mathrm{A}_{3}\mathrm{A}_{5}\mathrm{A}_{7}=\frac{1}{1296}

and

\mathrm{A}_{2}+\mathrm{A}_{4}=\frac{7}{36},

then, the value of

\mathrm{A}_{6}+\mathrm{A}_{8}+\mathrm{A}_{10}

is equal to

(A)33
(B)37
(C)43
(D)47

Q171·MathematicsNumericalJEE Main 2022

Let for

n = 1, 2, \ldots, 50

,

S_n

be the sum of the infinite geometric progression whose first term is

n^2

and whose common ratio is

\frac{1}{(n+1)^2}

. Then the value of

\frac{1}{26} + \sum\limits_{n=1}^{50}\left(S_n + \frac{2}{n+1} - n - 1\right)

is equal to

Q172·MathematicsSingle correctJEE Main 2022

If n arithmetic means are inserted between a and 100 such that the ratio of the first mean to the last mean is

1 : 7

and

a + n = 33

, then the value of n is

(A)21
(B)22
(C)23
(D)24

Q173·MathematicsNumericalJEE Main 2022

Let

A = \{1,\, a_1,\, a_2 \ldots\ldots a_{18},\, 77\}

be a set of integers with

1 < a_1 < a_2 < \ldots.. < a_{18} < 77

. Let the set

A + A = \{x + y : x, y \in A\}

contain exactly 39 elements. Then, the value of

a_1 + a_2 + \ldots.. + a_{18}

is equal to ______.

Q174·MathematicsSingle correctJEE Main 2022

Let

S=2+\dfrac{6}{7}+\dfrac{12}{7^{2}}+\dfrac{20}{7^{3}}+\dfrac{30}{7^{4}}+.....

then

4S

is equal to

(A) $\left(\dfrac{7}{3}\right)^{2}$
(B) $\dfrac{7^{3}}{3^{2}}$
(C) $\left(\dfrac{7}{3}\right)^{3}$
(D) $\dfrac{7^{2}}{3^{3}}$

Q175·MathematicsSingle correctJEE Main 2022

If

x = \sum_{n=0}^{\infty} a^{n}

,

y = \sum_{n=0}^{\infty} b^{n}

,

z = \sum_{n=0}^{\infty} c^{n}

, where a, b, c are in A.P. and

|a| < 1

,

|b| < 1

,

|c| < 1

,

abc \neq 0

, then

(A)x, y, z are in A.P.
(B)x, y, z are in G.P.
(C) $\frac{1}{x}, \frac{1}{y}, \frac{1}{z}$ are in A.P.
(D) $\frac{1}{x} + \frac{1}{y} + \frac{1}{z} = 1 - \left(a + b + c\right)$

Q176·MathematicsSingle correctJEE Main 2022

If

a_{1},a_{2},a_{3}....

and

b_{1},b_{2},b_{3}....

are A.P. and

a_{1}=2

,

a_{10}=3

,

a_{1}b_{1}=1=a_{10}b_{10}

then

a_{4}b_{4}

is equal to

(A) $\dfrac{35}{27}$
(B) $1$
(C) $\dfrac{27}{28}$
(D) $\dfrac{28}{27}$

Q177·MathematicsNumericalJEE Main 2022

If the sum of the first ten terms of the series

\frac{1}{5}+\frac{2}{65}+\frac{3}{325}+\frac{4}{1025}+\frac{5}{2501}+....

is

\frac{m}{n}

, where m and n are co-prime numbers, then m + n is equal to __________.

Q178·MathematicsSingle correctJEE Main 2022

If

A=\sum_{n=1}^{\infty}\frac{1}{\left(3+(-1)^n\right)^n}

and

B=\sum_{n=1}^{\infty}\frac{(-1)^n}{\left(3+(-1)^n\right)^n}

, then

\frac{A}{B}

is equal to :

(A) $\frac{11}{9}$
(B)1
(C) $-\frac{11}{9}$
(D) $-\frac{11}{3}$

Q179·MathematicsNumericalJEE Main 2022

If

a_{1}(>0)

,

a_{2}

,

a_{3}

,

a_{4}

,

a_{5}

are in a G.P.,

a_{2}+a_{4}=2a_{3}+1

and

3a_{2}+a_{3}=2a_{4}

, then

a_{2}+a_{4}+2a_{5}

is equal to ____.

Q180·MathematicsNumericalJEE Main 2022

Let

A = \{n \in N : \text{H.C.F. }(n, 45) = 1\}

and Let

B = \{2k : k \in \{1, 2, ..., 100\}\}

. Then the sum of all the elements of

A \cap B

is ___________.

Q181·MathematicsNumericalJEE Main 2022

Let

A = \sum_{i=1}^{10}\sum_{j=1}^{10}\min\{i, j\}

and

B = \sum_{i=1}^{10}\sum_{j=1}^{10}\max\{i, j\}

. Then

A + B

is equal to ________.

Q182·MathematicsSingle correctJEE Main 2022

The sum

1 + 2 \cdot 3 + 3 \cdot 3^2 + ..... + 10 \cdot 3^9

is equal to

(A) $\frac{2 \cdot 3^{12} + 10}{4}$
(B) $\frac{19 \cdot 3^{10} + 1}{4}$
(C) $5 \cdot 3^{10} - 2$
(D) $\frac{9 \cdot 3^{10} + 1}{2}$

Q183·MathematicsNumericalJEE Main 2022

The greatest integer less than or equal to the sum of first 100 terms of the sequence

\frac{1}{3}, \frac{5}{9}, \frac{19}{27}, \frac{65}{81}, \ldots\ldots

is equal to

Q184·MathematicsSingle correctJEE Main 2022

If

\frac{1}{2 \cdot 3^{10}} + \frac{1}{2^2 \cdot 3^9} + \cdots \frac{1}{2^{10} \cdot 3} = \frac{K}{2^{10} \cdot 3^{10}}

, then the remainder when K is divided by 6 is

(A) $1$
(B) $2$
(C) $3$
(D) $5$

Q185·MathematicsSingle correctJEE Main 2022

Let

f : N \to R

be a function such that

f(x+y)=2f(x)f(y)

for natural numbers x and y. If

f(1) = 2

, then the value of α for which

\sum_{k=1}^{10} f(\alpha + k) = \frac{512}{3}(2^{20} - 1)

holds, is

(A) $2$
(B) $3$
(C) $4$
(D) $6$

Q186·MathematicsNumericalJEE Main 2022

For a natural number n, let

a_n = 19^n - 12^n

. Then, the value of

\frac{31\alpha_9 - \alpha_{10}}{57\alpha_8}

is

Q187·MathematicsSingle correctJEE Main 2022

Let

x, y > 0

. If

x^{3}y^{2} = 2^{15}

, then the least value of

3x + 2y

is

(A)30
(B)32
(C)36
(D)40

Q188·MathematicsNumericalJEE Main 2022

The sum of all the elements of the set

\{\alpha \in \{1, 2, \ldots, 100\} : \mathrm{HCF}\,(\alpha, 24) = 1\}

is ______.

Q189·MathematicsSingle correctJEE Main 2022

If

\{a_i\}_{i=1}^{n}

where n is an even integer , is an arithmetic progression with common difference 1, and

\sum_{i=1}^{n} a_i = 192

,

\sum_{i=1}^{n/2} a_{2i} = 120

, then n is equal to:

(A)48
(B)96
(C)92
(D)104

Q190·MathematicsSingle correctJEE Advanced 2021

Let

M = \left\{ (x, y) \in \mathbb{R} \times \mathbb{R} : x^2 + y^2 \le r^2 \right\},

where

r > 0

. Consider the geometric progression

a_n = \frac{1}{2^{n-1}}

,

n = 1, 2, 3, \ldots

. Let

S_0 = 0

and, for

n \ge 1

, let

S_n

denote the sum of the first n terms of this progression. For

n \ge 1

, let

C_n

denote the circle with center

(S_{n-1}, 0)

and radius

a_n

, and

D_n

denote the circle with center

(S_{n-1}, S_{n-1})

and radius

a_n

. Consider M with

r = \frac{(2^{199} - 1)\sqrt{2}}{2^{198}}

. The number of all those circles

D_n

that are inside M is

(A)198
(B)199
(C)200
(D)201

Q191·MathematicsSingle correctJEE Advanced 2021

Let

M = \left\{ (x, y) \in \mathbb{R} \times \mathbb{R} : x^2 + y^2 \le r^2 \right\},

where

r > 0

. Consider the geometric progression

a_n = \frac{1}{2^{n-1}}

,

n = 1, 2, 3, \ldots

. Let

S_0 = 0

and, for

n \ge 1

, let

S_n

denote the sum of the first n terms of this progression. For

n \ge 1

, let

C_n

denote the circle with center

(S_{n-1}, 0)

and radius

a_n

, and

D_n

denote the circle with center

(S_{n-1}, S_{n-1})

and radius

a_n

. Consider M with

r = \frac{1025}{513}

. Let k be the number of all those circles

C_n

that are inside M. Let

l

be the maximum possible number of circles among these k circles such that no two circles intersect. Then

(A) $k + 2l = 22$
(B) $2k + l = 26$
(C) $2k + 3l = 34$
(D) $3k + 2l = 40$

Q192·MathematicsSingle correctJEE Main 2021

Let

a_{1}, a_{2}, ......., a_{21}

be an AP such that

\sum_{n=1}^{20}\frac{1}{a_{n}a_{n+1}} = \frac{4}{9}

. If the sum of this AP is 189, then

a_{6}a_{16}

is equal to :

(A)57
(B)72
(C)48
(D)36

Q193·MathematicsSingle correctJEE Main 2021

Let

S_{n} = 1 \cdot (n - 1) + 2 \cdot (n - 2) + 3 \cdot (n - 3) + \dots + (n - 1) \cdot 1

, n ≥ 4. The sum

\sum_{n=4}^{\infty}\left(\frac{2S_{n}}{n!} - \frac{1}{(n-2)!}\right)

is equal to :

(A) $\frac{e-1}{3}$
(B) $\frac{e-2}{6}$
(C) $\frac{e}{3}$
(D) $\frac{e}{6}$

Q194·MathematicsNumericalJEE Main 2021

If

S = \frac{7}{5} + \frac{9}{5^{2}} + \frac{13}{5^{3}} + \frac{19}{5^{4}} + ....

, then 160 S is equal to ______.

Q195·MathematicsSingle correctJEE Main 2021

Three numbers are in an increasing geometric progression with common ratio r. If the middle number is doubled, then the new numbers are in an arithmetic progression with common difference d. If the fourth term of GP is

3r^{2}

, then

r^{2} - d

is equal to :

(A) $7 - 7\sqrt{3}$
(B) $7 + \sqrt{3}$
(C) $7 - \sqrt{3}$
(D) $7 + 3\sqrt{3}$

Q196·MathematicsSingle correctJEE Main 2021

The sum of 10 terms of the series

\frac{3}{1^{2} \times 2^{2}} + \frac{5}{2^{2} \times 3^{2}} + \frac{7}{3^{2} \times 4^{2}} + ....

is :

(A)1
(B) $\frac{120}{121}$
(C) $\frac{99}{100}$
(D) $\frac{143}{144}$

Q197·MathematicsSingle correctJEE Main 2021

Let

a_1, a_2, a_3, \ldots

be an A.P. If

\frac{a_1 + a_2 + \ldots + a_{10}}{a_1 + a_2 + \ldots + a_p} = \frac{100}{p^2}

,

p \neq 10

, then

\frac{a_{11}}{a_{10}}

is equal to :

(A) $\frac{19}{21}$
(B) $\frac{100}{121}$
(C) $\frac{21}{19}$
(D) $\frac{121}{100}$

Q198·MathematicsSingle correctJEE Main 2021

If 0 < x < 1 and

y = \frac{1}{2}x^{2} + \frac{2}{3}x^{3} + \frac{3}{4}x^{4} + \ldots

, then the value of

e^{1+y}

at

x = \frac{1}{2}

is:

(A) $\frac{1}{2}e^{2}$
(B)2e
(C) $\frac{1}{2}\sqrt{e}$
(D) $2e^{2}$

Q199·MathematicsSingle correctJEE Main 2021

If

0 < x < 1

, then

\frac{3}{2}x^2 + \frac{5}{3}x^3 + \frac{7}{4}x^4 + .....

, is equal to :

(A) $x\left(\frac{1+x}{1-x}\right) + \log_e(1-x)$
(B) $x\left(\frac{1-x}{1+x}\right) + \log_e(1-x)$
(C) $\frac{1-x}{1+x} + \log_e(1-x)$
(D) $\frac{1+x}{1-x} + \log_e(1-x)$

Q200·MathematicsSingle correctJEE Main 2021

If for

x, y \in \mathbf{R}

,

x > 0

,

y = \log_{10}x + \log_{10}x^{1/3} + \log_{10}x^{1/9} + .....

upto

\infty

terms and

\frac{2+4+6+....+2y}{3+6+9+....+3y} = \frac{4}{\log_{10} x}

, then the ordered pair

(x, y)

is equal to :

(A) $(10^6, 6)$
(B) $(10^4, 6)$
(C) $(10^2, 3)$
(D) $(10^6, 9)$

Q201·MathematicsSingle correctJEE Main 2021

If the sum of an infinite GP a, ar,

ar^2

,

ar^3

,... is 15 and the sum of the squares of its each term is 150, then the sum of

ar^2

,

ar^4

,

ar^6

, ... is :

(A) $\frac{5}{2}$
(B) $\frac{1}{2}$
(C) $\frac{25}{2}$
(D) $\frac{9}{2}$

Q202·MathematicsNumericalJEE Main 2021

Let

a_{1}, a_{2}, ....., a_{10}

be an AP with common difference

-3

and

b_{1}, b_{2}, ....., b_{10}

be a GP with common ratio 2. Let

c_{k} = a_{k} + b_{k}

,

k = 1, 2, ..., 10

. If

c_{2} = 12

and

c_{3} = 13

, then

\sum_{k=1}^{10} c_{k}

is equal to ________.

Q203·MathematicsNumericalJEE Main 2021

If

\log_3 2, \log_3\left(2^x-5\right), \log_3\left(2^x-\frac{7}{2}\right)

are in an arithmetic progression, then the value of x is equal to ____.

Q204·MathematicsSingle correctJEE Main 2021

Let

S_{n}

be the sum of the first n terms of an arithmetic progression. If

S_{3n}=3S_{2n}

, then the value of

\frac{S_{4n}}{S_{2n}}

is ;

(A)2
(B)6
(C)8
(D)4

Q205·MathematicsNumericalJEE Main 2021

If the value of

\left( 1 + \dfrac{2}{3} + \dfrac{6}{3^{2}} + \dfrac{10}{3^{3}} + .... \text{upto } \infty \right)^{\log_{0.25}\left( \dfrac{1}{3} + \dfrac{1}{3^{2}} + .... \text{upto } \infty \right)}

is l, then

l^{2}

is equal to ........

Q206·MathematicsSingle correctJEE Main 2021

Let

S_n

denote the sum of the first n-terms of an arithmetic progression. If

S_{10} = 530

,

S_5 = 140

, then

S_{20} - S_6

is equal to :

(A) $1852$
(B) $1842$
(C) $1872$
(D) $1862$

Q207·MathematicsNumericalJEE Main 2021

Let

\{a_{n}\}_{n=1}^{\infty}

be a sequence such that

a_{1}=1, a_{2}=1

and

a_{n+2}=2a_{n+1}+a_{n}

for all

n\ge 1

. Then the value of

47\sum_{n=1}^{\infty}\frac{a_{n}}{2^{3n}}

is equal to……..

Q208·MathematicsSingle correctJEE Main 2021

If sum of the first 21 terms of the series

\log_{9^{1/2}}x+\log_{9^{1/3}}x+\log_{9^{1/4}}x+....,

where

x>0

is 504, then x is equal to :

(A) $7$
(B) $9$
(C) $243$
(D) $81$

Q209·MathematicsNumericalJEE Main 2021

If

\sum_{r=1}^{10} r!\left(r^{3} + 6r^{2} + 2r + 5\right) = \alpha(11!)

, then the value of α is equal to ________ .

Q210·MathematicsSingle correctJEE Main 2021

Let

S_{1}

be the sum of first 2n terms of an arithmetic progression. Let

S_{2}

be the sum of first 4n terms of the same arithmetic progression. If

(S_{2} - S_{1})

is 1000, then the sum of the first 6n terms of the arithmetic progression is equal to:

(A)1000
(B)7000
(C)5000
(D)3000

Q211·MathematicsSingle correctJEE Main 2021

\dfrac{1}{3^{2}-1} + \dfrac{1}{5^{2}-1} + \dfrac{1}{7^{2}-1} + ... + \dfrac{1}{\left(201\right)^{2}-1}

is equal to

(A) $\dfrac{101}{404}$
(B) $\dfrac{25}{101}$
(C) $\dfrac{101}{408}$
(D) $\dfrac{99}{400}$

Q212·MathematicsNumericalJEE Main 2021

The missing value in the following figure is

Q213·MathematicsSingle correctJEE Main 2021

If

\alpha, \beta

are natural numbers such that

100^{\alpha} - 199\beta = (100)(100) + (99)(101) + (98)(102) + ..... + (1)(199)

, then the slope of the line passing through

(\alpha, \beta)

and origin is :

(A) $540$
(B) $550$
(C) $530$
(D) $510$

Q214·MathematicsSingle correctJEE Main 2021

The value of

4 + \cfrac{1}{5 + \cfrac{1}{4 + \cfrac{1}{5 + \cfrac{1}{4 + .......\infty}}}}

is :

(A) $2 + \frac{2}{5}\sqrt{30}$
(B) $2 + \frac{4}{\sqrt{5}}\sqrt{30}$
(C) $4 + \frac{4}{\sqrt{5}}\sqrt{30}$
(D) $5 + \frac{2}{5}\sqrt{30}$

Q215·MathematicsNumericalJEE Main 2021

Consider an arithmetic series and a geometric series having four initial terms from the set {11, 8, 21, 16, 26, 32, 4}. If the last terms of these series are the maximum possible four digit numbers, then the number of common terms in these two series is equal to ______ .

Q216·MathematicsNumericalJEE Main 2021

Let

S_{n}(x)=\log_{a^{1/2}}x+\log_{a^{1/3}}x+\log_{a^{1/6}}x

+\log_{a^{1/11}}x+\log_{a^{1/18}}x+\log_{a^{1/27}}x+.....

up to n-terms, where a > 1. If

S_{24}(x) = 1093

and

S_{12}(2x) = 265

, then value of a is equal to _______.

Q217·MathematicsNumericalJEE Main 2021

Let

\frac{1}{16}

, a and b be in G.P. and

\frac{1}{a},\frac{1}{b},6

be in A.P., where a, b > 0. Then

72(a + b)

is equal to _________.

Q218·MathematicsSingle correctJEE Main 2021

The sum of the infinite series

1 + \frac{2}{3} + \frac{7}{3^{2}} + \frac{12}{3^{3}} + \frac{17}{3^{4}} + \frac{22}{3^{5}} + \ldots

is equal to

(A) $\frac{9}{4}$
(B) $\frac{15}{4}$
(C) $\frac{13}{4}$
(D) $\frac{11}{4}$

Q219·MathematicsSingle correctJEE Main 2021

In an increasing, geometric series, the sum of the second and the sixth term is

\frac{25}{2}

and the product of the third and fifth term is 25. Then, the sum of

4^{th}

,

6^{th}

and

8^{th}

terms is equal to :

(A)35
(B)30
(C)26
(D)32

Q220·MathematicsSingle correctJEE Main 2021

The sum of the series

\sum_{n=1}^{\infty} \frac{n^{2} + 6n + 10}{(2n+1)!}

is equal to :

(A) $\frac{41}{8} e + \frac{19}{8} e^{-1} - 10$
(B) $-\frac{41}{8} e + \frac{19}{8} e^{-1} - 10$
(C) $\frac{41}{8} e - \frac{19}{8} e^{-1} - 10$
(D) $\frac{41}{8} e + \frac{19}{8} e^{-1} + 10$

Q221·MathematicsNumericalJEE Main 2021

If the arithmetic mean and geometric mean of the

p^{th}

and

q^{th}

terms of the sequence

-16

, 8,

-4

, 2, ………… satisfy the equation

4x^{2} - 9x + 5 = 0

, then p+q is equal to ________________.

Q222·MathematicsNumericalJEE Main 2021

Let

A_1, A_2, A_3, \dots

be squares such that for each

n \ge 1

, the length of the side of

A_n

equals the length of diagonal of

A_{n+1}

. If the length of

A_1

is 12 cm, then the smallest value of

n

for which area of

A_n

is less than one, is __________.

Q223·MathematicsSingle correctJEE Main 2021

If

0 < \theta, \phi < \frac{\pi}{2}

,

x = \sum_{n=0}^{\infty} \cos^{2n}\theta

,

y = \sum_{n=0}^{\infty} \sin^{2n}\phi

and

z = \sum_{n=0}^{\infty} \cos^{2n}\theta \cdot \sin^{2n}\phi

then :

(A) $xyz = 4$
(B) $xy - z = (x + y)z$
(C) $xy + yz + zx = z$
(D) $xy + z = (x + y)z$

Q224·MathematicsNumericalJEE Main 2021

The sum of first four terms of a geometric progression (G.P.) is

\frac{65}{12}

and the sum of their respective reciprocals is

\frac{65}{18}

. If the product of first three terms of the G.P. is 1, and the third term is

\alpha

, then

2\alpha

is__________.

Q225·MathematicsNumericalJEE Main 2021

Let

A = \{n \in N : n \text{ is a 3-digit number}\}

B = \{9k + 2 : k \in N\}

and

C : \{9k + \ell : k \in N\}

for some

\ell

(0 < \ell < 9)

If the sum of all the elements of the set

A \cap (B \cup C)

is

274 \times 400

, then

\ell

is equal to __

Q226·MathematicsNumericalJEE Advanced 2020

Let

a_{1}

,

a_{2}

,

a_{3}

, ..... be a sequence of positive integers in arithmetic progression with common difference 2. Also, let

b_{1}

,

b_{2}

,

b_{3}

, ..... be a sequence of positive integers in geometric progression with common ratio 2. If

a_{1} = b_{1} = c

, then the number of all possible values of c, for which the equality

2\left(a_{1} + a_{2} + \ldots + a_{n}\right) = b_{1} + b_{2} + \ldots + b_{n}

holds for some positive integer n, is ______

Q227·MathematicsNumericalJEE Advanced 2020

Let m be the minimum possible value of

\log_{3}\left(3^{y_{1}} + 3^{y_{2}} + 3^{y_{3}}\right)

, where

y_{1}, y_{2}, y_{3}

are real numbers for which

y_{1} + y_{2} + y_{3} = 9

. Let M be the maximum possible value of

\left(\log_{3} x_{1} + \log_{3} x_{2} + \log_{3} x_{3}\right)

, where

x_{1}, x_{2}, x_{3}

are positive real numbers for which

x_{1} + x_{2} + x_{3} = 9

. Then the value of

\log_{2}\left(m^{3}\right) + \log_{3}\left(M^{2}\right)

is ______.

Q228·MathematicsSingle correctJEE Main 2020

Let a, b, c, d and p be any non zero distinct real numbers such that

(a^{2}+b^{2}+c^{2})p^{2}-2(ab+bc+cd)p+(b^{2}+c^{2}+d^{2})=0

. Then:

(A)a, c, p are in A.P.
(B)a, c, p are in G.P.
(C)a, b, c, d are in G.P.
(D)a, b, c, d are in A.P.

Q229·MathematicsSingle correctJEE Main 2020

If

f(x+y) = f(x)f(y)

and

\displaystyle\sum_{x=1}^{\infty} f(x) = 2,\ x, y \in N

where N is the set of all natural numbers, then the value of

\dfrac{f(4)}{f(2)}

is:

(A) $\dfrac{2}{3}$
(B) $\dfrac{1}{9}$
(C) $\dfrac{1}{3}$
(D) $\dfrac{4}{9}$

Q230·MathematicsSingle correctJEE Main 2020

The common difference of the A.P.

b_{1}, b_{2}, \ldots, b_{m}

is 2 more than the common difference of A.P.

a_{1}, a_{2}, \ldots, a_{n}

. If

a_{40} = -159

,

a_{100} = -399

and

b_{100} = a_{70}

, then

b_{1}

is equal to:

(A)81
(B)−127
(C)−81
(D)127

Q231·MathematicsSingle correctJEE Main 2020

If

2^{10} + 2^{9}.3^{1} + 2^{8}.3^{2} + ... + 2.3^{9} + 3^{10} = S - 2^{11}

, then S is equal to

(A) $\dfrac{3^{11}}{2} + 2^{10}$
(B) $3^{11} - 2^{12}$
(C) $2.3^{11}$
(D) $3^{11}$

Q232·MathematicsSingle correctJEE Main 2020

If the sum of the first 20 terms of the series

\log_{(7^{1/2})} x + \log_{(7^{1/3})} x + \log_{(7^{1/4})} x + ...

is 460, then x is equal to:

(A) $7^2$
(B) $7^{46/21}$
(C) $7^{1/2}$
(D) $e^2$

Q233·MathematicsSingle correctJEE Main 2020

If the sum of the second, third and fourth terms of a positive term G.P. is 3 and the sum of its sixth, seventh and eights terms is 243, then the sum of the first 50 terms of this G.P. is:

(A) $\dfrac{1}{26}\left(3^{50}-1\right)$
(B) $\dfrac{1}{13}\left(3^{50}-1\right)$
(C) $\dfrac{2}{13}\left(3^{50}-1\right)$
(D) $\dfrac{1}{26}\left(3^{49}-1\right)$

Q234·MathematicsSingle correctJEE Main 2020

If

3^{2\sin 2\alpha - 1}

, 14 and

3^{4-2\sin 2\alpha}

are the first three terms of an AP for some

\alpha

, then the sixth term of this AP is:

(A)65
(B)78
(C)66
(D)81

Q235·MathematicsSingle correctJEE Main 2020

If

1 + (1 - 2^{2} \cdot 1) + (1 - 4^{2} \cdot 3) + (1 - 6^{2} \cdot 5) + \ldots\ldots + (1 - 20^{2} \cdot 19) = \alpha - 220\beta

, then an ordered pair

(\alpha, \beta)

is equal to:

(A) $(11, 97)$
(B) $(10, 97)$
(C) $(10, 103)$
(D) $(11, 103)$

Q236·MathematicsSingle correctJEE Main 2020

If the first term of an A.P. is 3 and the sum of its first 25 terms is equal to the sum of its next 15 terms, then the common difference of this A.P. is:

(A) $\frac{1}{4}$
(B) $\frac{1}{7}$
(C) $\frac{1}{6}$
(D) $\frac{1}{5}$

Q237·MathematicsNumericalJEE Main 2020

If m arithmetic means (A.Ms) and three geometric means (G.Ms) are inserted between 3 and 243 such that

4^{th}

A.M. is equal to

2^{nd}

G.M., then m is equal to ________________.

Q238·MathematicsNumericalJEE Main 2020

The value of

(0.16)^{\log_{2.5}\left(\frac{1}{3} + \frac{1}{3^{2}} + \frac{1}{3^{3}} + \ldots \text{ to } \infty\right)}

is equal to __________.

Q239·MathematicsSingle correctJEE Main 2020

If the sum of the series

20 + 19\frac{3}{5} + 19\frac{1}{5} + 18\frac{4}{5} + ....

upto

n^{th}

term is 488 and the

n^{th}

term is negative, then:

(A) $n^{th}$ term is $-4\frac{2}{5}$
(B) $n = 41$
(C) $n^{th}$ term is $-4$
(D) $n = 60$

Q240·MathematicsNumericalJEE Main 2020

The number of terms common to the two A.P.'s 3, 7, 11,…………..407 and 2, 9, 16, …..709 is ______

Q241·MathematicsSingle correctJEE Main 2020

The product

2^{\frac{1}{4}}.4^{\frac{1}{16}}.8^{\frac{1}{48}}.16^{\frac{1}{128}}

.......... to

\infty

is equal to

(A) $2^{\frac{1}{2}}$
(B)2
(C)1
(D) $2^{\frac{1}{4}}$

Q242·MathematicsSingle correctJEE Main 2020

If

x = \sum\limits_{n=0}^{\infty} (-1)^{n} \tan^{2n}\theta

and

y = \sum\limits_{n=0}^{\infty} \cos^{2n}\theta

, for

0 < \theta < \dfrac{\pi}{4}

, then:

(A) $y(1 - x) = 1$
(B) $y(1 + x) = 1$
(C) $x(1 + y) = 1$
(D) $x(1 - y) = 1$

Q243·MathematicsSingle correctJEE Main 2020

Let

a_{n}

be the

n^{th}

term of a G.P. of positive terms. If

\sum\limits_{n=1}^{100} a_{2n+1} = 200

and

\sum\limits_{n=1}^{100} a_{2n} = 100

, then

\sum\limits_{n=1}^{200} a_{n}

is equal to:

(A)225
(B)300
(C)150
(D)175

Q244·MathematicsSingle correctJEE Main 2020

Let

f:R\to R

be such that for all

x\in R

,

\left(2^{1+x}+2^{1-x}\right), f(x)

and

\left(3^{x}+3^{-x}\right)

are in A.P., then the minimum value of

f(x)

is:

(A) $3$
(B) $4$
(C) $2$
(D) $0$

Q245·MathematicsSingle correctJEE Main 2020

If the

10^{th}

term of an A.P. is

\dfrac{1}{20}

and its

20^{th}

term is

\dfrac{1}{10}

, then the sum of its first 200 terms is:

(A)100
(B) $50\dfrac{1}{4}$
(C) $100\dfrac{1}{2}$
(D)50

Q246·MathematicsNumericalJEE Main 2020

The sum

\displaystyle\sum_{k=1}^{20}(1+2+3+....+k)

is

Q247·MathematicsNumericalJEE Main 2020

The sum

\sum_{n=1}^{7}\dfrac{n(n+1)(2n+1)}{4}

is equal to ___________.

Q248·MathematicsSingle correctJEE Main 2020

If the sum of the first 40 terms of the series,

3+4+8+9+13+14+18+19+....

is (102)m, then m is equal to:

(A)10
(B)5
(C)20
(D)25

Q249·MathematicsSingle correctJEE Main 2020

Let

a_1,a_2,a_3,........

be a G.P. such that

a_1<0, a_1+a_2=4

and

a_3+a_4=16

. If

\displaystyle\sum_{i=1}^{9}a_i=4\lambda

, then

\lambda

is equal to:

(A)-171
(B)-513
(C)171
(D) $\dfrac{511}{3}$

Q250·MathematicsSingle correctJEE Main 2020

Five numbers are in A.P., whose sum is 25 and product is 2520. If one of these five numbers is

-\frac{1}{2}

, then the greatest number amongst them is:

(A)7
(B) $\frac{21}{2}$
(C)16
(D)27

Q251·MathematicsSingle correctJEE Main 2020

The greatest positive integer k, for which

49^{k}+1

is a factor of the sum

49^{125} + 49^{124} + \ldots\ldots + 49^{2} + 49 + 1

, is:

(A)65
(B)63
(C)32
(D)60

Q252·MathematicsMultiple correctJEE Advanced 2019

Let

\alpha

and

\beta

be the roots of

x^2 - x - 1 = 0

, with

\alpha > \beta

. For all positive integers n, define

a_n = \frac{\alpha^n - \beta^n}{\alpha - \beta}

,

n \ge 1

,

b_1 = 1

and

b_n = a_{n-1} + a_{n+1}

,

n \ge 2

. Then which of the following options is/are correct?

(A) $\sum_{n=1}^{\infty} \frac{b_n}{10^n} = \frac{8}{89}$
(B) $b_n = \alpha^n + \beta^n$ for all $n \ge 1$
(C) $a_1 + a_2 + a_3 + ..... + a_n = a_{n+2} - 1$ for all $n \ge 1$
(D) $\sum_{n=1}^{\infty} \frac{a_n}{10^n} = \frac{10}{89}$

Q253·MathematicsNumericalJEE Advanced 2019

Let AP(a; d) denote the set of all the terms of an infinite arithmetic progression with first term a and common difference

d > 0

. If

AP(1; 3) \cap AP(2; 5) \cap AP(3; 7) = AP(a; d)

then

a + d

equals ____

Q254·MathematicsSingle correctJEE Main 2019

Let S

_{n}

denote the sum of the first n terms of an A.P.. If S

_{4}

= 16 and S

_{6}

=

-48

, then S

_{10}

is equal to :

(A) $-410$
(B) $-260$
(C) $-320$
(D) $-380$

Q255·MathematicsSingle correctJEE Main 2019

If

a_{1}, a_{2}, a_{3}, \ldots\ldots

are in A.P. such that

a_{1} + a_{7} + a_{16} = 40

, then the sum of the first 15 terms of this A.P. is:

(A)200
(B)280
(C)150
(D)120

Q256·MathematicsSingle correctJEE Main 2019

If

\alpha

and

\beta

are the roots of the equation

375x^{2}-25x-2=0

, then

\lim_{n \to \infty}\sum_{r=1}^{n}\alpha^{r} + \lim_{n \to \infty}\sum_{r=1}^{n}\beta^{r}

is equal to :

(A) $\frac{1}{12}$
(B) $\frac{29}{358}$
(C) $\frac{7}{116}$
(D) $\frac{21}{346}$

Q257·MathematicsSingle correctJEE Main 2019

The sum

1 + \dfrac{1^{3} + 2^{3}}{1 + 2} + \dfrac{1^{3} + 2^{3} + 3^{3}}{1 + 2 + 3} + \ldots + \dfrac{1^{3} + 2^{3} + 3^{3} + \ldots + 15^{3}}{1 + 2 + 3 + \ldots + 15} - \dfrac{1}{2}\left(1 + 2 + 3 + \ldots + 15\right)

is equal to

(A)620
(B)1860
(C)1240
(D)660

Q258·MathematicsSingle correctJEE Main 2019

Let

a_1

,

a_2

,

a_3

, …… be and A.P with

a_6

= 2. Then the common difference of this A.P., which maximizes the product

a_1a_4a_5

is

(A) $\dfrac{3}{2}$
(B) $\dfrac{8}{5}$
(C) $\dfrac{2}{3}$
(D) $\dfrac{6}{5}$

Q259·MathematicsSingle correctJEE Main 2019

The sum

\frac{3 \times 1}{1^{2}} + \frac{5 \times (1^{3}+2^{3})}{1^{2}+2^{2}} + \frac{7 \times (1^{3}+2^{3}+3^{3})}{1^{2}+2^{2}+3^{2}} + ........

upto 10

^{th}

term, is

(A)620
(B)660
(C)680
(D)600

Q260·MathematicsSingle correctJEE Main 2019

Let a, b and c be in G.P with common ratio r, where

a \neq 0

and

0 < r \leq \dfrac{1}{2}

. If 3a, 7b and 15c are the first three terms of an A.P., then the

4^{\text{th}}

term of this A.P is

(A) $\dfrac{2}{3}a$
(B) $\dfrac{7}{3}a$
(C) $5a$
(D) $a$

Q261·MathematicsSingle correctJEE Main 2019

If a

_{1}

, a

_{2}

, a

_{3}

........... a

_{n}

are in A.P and a

_{1}

+ a

_{4}

+ a

_{7}

+ ............... + a

_{16}

= 114, then a

_{1}

+ a

_{6}

+ a

_{11}

+ a

_{16}

is equal to

(A)76
(B)64
(C)98
(D)38

Q262·MathematicsSingle correctJEE Main 2019

Some identical balls are arranged in rows to form an equilateral triangle. The first row consists of one ball, the second row consists of two balls and so on. If 99 more identical balls are added to the total number of balls used in forming the equilateral triangle, then all these balls can be arranged in a square whose each side contains exactly 2 balls less than the number of balls each side of the triangle contains. Then the number of balls used to form the equilateral triangle is

(A)190
(B)262
(C)225
(D)157

Q263·MathematicsSingle correctJEE Main 2019

The sum of the series

1+2\times3+3\times5+4\times7+....

upto

11^{th}

term is

(A)915
(B)946
(C)945
(D)916

Q264·MathematicsSingle correctJEE Main 2019

Let

\displaystyle\sum_{k=1}^{10} f(a+k)=16(2^{10}-1)

, where the function f satisfies

f(x+y)=f(x)f(y)

for all natural numbers x, y and

f(1)=2

. Then the natural number 'a' is:

(A)4
(B)16
(C)2
(D)3

Q265·MathematicsSingle correctJEE Main 2019

If the sum and product of the first three term in an A.P. are 33 and 1155, respectively, then a value of its 11

^{th}

tern is

(A) $-25$
(B) $25$
(C) $-36$
(D) $-35$

Q266·MathematicsSingle correctJEE Main 2019

Let the sum of the first n terms of a non-constant A.P.,

a_1, a_2, a_3, ......

be

50n + \dfrac{n(n-7)}{2}A

, where A is a constant. If d is the common difference of this A.P., then the ordered pair

(d, a_{50})

is equal to:

(A) $(A, 50+46A)$
(B) $(A, 50+45A)$
(C) $(50, 50+45A)$
(D) $(50, 50+46A)$

Q267·MathematicsSingle correctJEE Main 2019

Then sum

\sum_{k=1}^{20}k\frac{1}{2^{k}}

is equal to:

(A) $2-\frac{11}{2^{19}}$
(B) $1-\frac{11}{2^{20}}$
(C) $2-\frac{21}{2^{20}}$
(D) $2-\frac{3}{2^{17}}$

Q268·MathematicsSingle correctJEE Main 2019

The sum of all natural numbers 'n' such that

100<n<200

and H.C. F (91, n) > 1 is:

(A)3221
(B)3303
(C)3203
(D)3121

Q269·MathematicsSingle correctJEE Main 2019

Let

S_{k}=\frac{1+2+3+.....+k}{k}

. If

S_{1}^{2}+S_{2}^{2}+.....+S_{10}^{2}=\frac{5}{12}A

, then A equal to:

(A)283
(B)301
(C)303
(D)156

Q270·MathematicsSingle correctJEE Main 2019

The product of three consecutive terms of a G.P. is 512. If 4 is added to each of the first and the second of these terms, the three terms now form an A.P. Then the sum of the original three terms of the given G.P. is:

(A)36
(B)32
(C)24
(D)28

Q271·MathematicsSingle correctJEE Main 2019

If the sum of the first 15 terms of the series

\left(\frac{3}{4}\right)^{3} + \left(1\frac{1}{2}\right)^{3} + \left(2\frac{1}{4}\right)^{3} + 3^{3} + \left(3\frac{3}{4}\right)^{3} + .....

is equal to 225 k, then k is equal to :

(A)108
(B)27
(C)54
(D)9

Q272·MathematicsSingle correctJEE Main 2019

Let

a_{1}, a_{2}, \ldots, a_{10}

be a G.P. If

\frac{a_{3}}{a_{1}}=25

, then

\frac{a_{9}}{a_{5}}

equals:

(A) $5^{4}$
(B) $4(5^{2})$
(C) $5^{3}$
(D) $2(5^{2})$

Q273·MathematicsSingle correctJEE Main 2019

If

19^{th}

terms of non – zero A.P. is zero, then its (

49^{th}

term) : (

29^{th}

term) is:

(A)4:1
(B)1:3
(C)3:1
(D)2:1

Q274·MathematicsSingle correctJEE Main 2019

The sum of an infinite geometric series with positive terms is 3 and the sum of the cubes of its terms is

\frac{27}{19}

. Then the common ratio of this series is:

(A) $\frac{1}{3}$
(B) $\frac{2}{3}$
(C) $\frac{2}{9}$
(D) $\frac{4}{9}$

Q275·MathematicsSingle correctJEE Main 2019

The sum of all two digit positive numbers which when divided by 7 yield 2 or 5 as remainder is:

(A)1256
(B)1465
(C)1365
(D)1356

Q276·MathematicsSingle correctJEE Main 2019

If 5, 5r, 5r

^{2}

are the lengths of the sides of a triangle, then r cannot be equal to:

(A) $\frac{3}{4}$
(B) $\frac{5}{4}$
(C) $\frac{7}{4}$
(D) $\frac{3}{2}$

Q277·MathematicsSingle correctJEE Main 2019

The sum of the following series

1+6+\frac{9(1^{2}+2^{2}+3^{2})}{7}+\frac{12(1^{2}+2^{2}+3^{2}+4^{2})}{9}+\frac{15(1^{2}+2^{2}+....+5^{2})}{11}+...

up to 15 terms, is:

(A)7820
(B)7830
(C)7520
(D)7510

Q278·MathematicsSingle correctJEE Main 2019

Let a, b and c be the

7^{th}

,

11^{th}

and

13^{th}

terms respectively of a non

-

constant A.P. If these are also the three consecutive terms of a G.P. then

\frac{a}{c}

is equal to:

(A) $\frac{1}{2}$
(B)4
(C)2
(D) $\frac{7}{13}$

Q279·MathematicsSingle correctJEE Main 2019

If a, b and c be three distinct numbers in G.P. and

a + b + c = xb

then x can not be

(A) $-2$
(B) $-3$
(C) $4$
(D) $2$

Q280·MathematicsNumericalJEE Advanced 2018

Let X be the set consisting of the first 2018 terms of the arithmetic progression 1, 6, 11, ….. , and Y be the set consisting of the first 2018 terms of arithmetic progression 9, 16, 23, ….. . Then, the number of elements in the set

X \cup Y

is ______ .

Q281·MathematicsIntegerJEE Advanced 2017

The sides of a right angled triangle are in arithmetic progression. If the triangle has area 24, then what is the length of its smallest side ?

Q282·MathematicsSingle correctJEE Advanced 2017

PARAGRAPH 2 Let

p

,

q

be integers and let

\alpha

,

\beta

be the roots of the equation

x^{2} - x - 1 = 0

, where

\alpha \ne \beta

. For

n = 0, 1, 2, \dots

, let

a_{n} = p\alpha^{n} + q\beta^{n}

. FACT: If

a

and

b

are rational numbers and

a + b\sqrt{5} = 0

, then

a = 0 = b

.

a_{12} =

(A) $a_{11} - a_{10}$
(B) $a_{11} + a_{10}$
(C) $2a_{11} + a_{10}$
(D) $a_{11} + 2a_{10}$

Q283·MathematicsSingle correctJEE Advanced 2016

Let

b_{i} > 1

for

i = 1, 2, \ldots, 101

. Suppose

\log_{e} b_{1}, \log_{e} b_{2}, \ldots, \log_{e} b_{101}

are in Arithmetic Progression (A. P.) with the common difference

\log_{e} 2

. Suppose

a_{1}, a_{2}, \ldots, a_{101}

are in A.P. such that

a_{1} = b_{1}

and

a_{51} = b_{51}

. If

t = b_{1} + b_{2} + \ldots + b_{51}

and

s = a_{1} + a_{2} + \ldots + a_{51}

, then

(A) $s > t$ and $a_{101} > b_{101}$
(B) $s > t$ and $a_{101} < b_{101}$
(C) $s < t$ and $a_{101} > b_{101}$
(D) $s < t$ and $a_{101} < b_{101}$

Q284·MathematicsIntegerJEE Advanced 2015

Suppose that all the terms of an arithmetic progression (A.P.) are natural numbers. If the ratio of the sum of the first seven terms to the sum of the first eleven terms is 6 : 11 and the seventh term lies in between 130 and 140, then the common difference of this A.P. is

Q285·MathematicsIntegerJEE Advanced 2014

Let

a, b, c

be positive integers such that

\frac{b}{a}

is an integer. If

a, b, c

are in geometric progression and the arithmetic mean of

a, b, c

is

b + 2

, then the value of

\frac{a^2 + a - 14}{a+1}

is __________

Q286·MathematicsIntegerJEE Advanced 2013

A pack contains n cards numbered from 1 to n. Two consecutive numbered cards are removed from the pack and the sum of the numbers on the remaining cards is 1224. If the smaller of the numbers on the removed cards is k, then

k-20=

________

Q287·MathematicsMultiple correctJEE Advanced 2013

Let

S_n=\sum_{k=1}^{4n}(-1)^{\frac{k(k+1)}{2}}k^2

. Then

S_n

can take value(s)

(A) $1056$
(B) $1088$
(C) $1120$
(D) $1332$

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