Sequences & Series

31 JEE Main questions · Maths

MathsJEE Main 21 Jan 2026, Shift 1·Q6·MCQ·Moderate
Open
Geometric progression
Let $a_1, a_2, a_3, \ldots$ be a G.P. of increasing positive terms such that $a_2\cdot a_3\cdot a_4 = 64$ and $a_1 + a_3 + a_5 = \dfrac{813}{7}$ . Then $a_3 + a_5 + a_7$ is equal to:
1. (1)3256
2. (2)3252
3. (3)3244
4. (4)3248
MathsJEE Main 21 Jan 2026, Shift 1·Q22·Numerical·Tough
Open
Recursive sequence and series sum
Let $a_1 = 1$ and for $n \ge 1$ , $a_{n+1} = \dfrac12 a_n + \dfrac{n^2 - 2n - 1}{n^2(n+1)^2}$ . Then $\left\lvert\displaystyle\sum_{n=1}^{\infty}\left(a_n - \frac{2}{n^2}\right)\right\rvert$ is equal to .
MathsJEE Main 21 Jan 2026, Shift 2·Q1·MCQ·Moderate
Open
Sum of a finite series
The positive integer $n$ , for which the solutions of the equation $x(x+2) + (x+2)(x+4) + \ldots + (x+2n-2)(x+2n) = \dfrac{8n}{3}$ are two consecutive even integers, is:
1. (1)3
2. (2)6
3. (3)12
4. (4)9
MathsJEE Main 21 Jan 2026, Shift 2·Q12·MCQ·Easy–moderate
Open
Geometric progression
Let $a_1, \dfrac{a_2}{2}, \dfrac{a_3}{2^2}, \ldots, \dfrac{a_{10}}{2^9}$ be a G.P. of common ratio $\dfrac{1}{\sqrt2}$ . If $a_1 + a_2 + \ldots + a_{10} = 62$ , then $a_1$ is equal to:
1. (1) $2\left(\sqrt2 - 1\right)$
2. (2) $2 - \sqrt2$
3. (3) $\sqrt2 - 1$
4. (4) $2(2 - \sqrt2)$
MathsJEE Main 22 Jan 2026, Shift 1·Q20·MCQ·Easy–moderate
Open
Arithmetic progression sums
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
1. (1) $-20$
2. (2) $-24$
3. (3) $-26$
4. (4) $-22$
MathsJEE Main 22 Jan 2026, Shift 2·Q24·Numerical·Moderate
Open
Arithmetic and geometric progressions
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 .
MathsJEE Main 23 Jan 2026, Shift 2·Q20·MCQ·Easy–moderate
Open
Nth term from sum
Let $\displaystyle\sum_{k=1}^{n} a_k = \alpha n^2 + \beta n$ . If $a_{10} = 59$ and $a_6 = 7a_1$ then $\alpha + \beta$ is equal to
1. (1)12
2. (2)3
3. (3)5
4. (4)7
MathsJEE Main 24 Jan 2026, Shift 1·Q4·MCQ·Moderate
Open
Geometric progression sum
Let $729, 81, 9, 1, \ldots$ be a sequence and $P_n$ denote the product of the first $n$ terms of this sequence. If $2\sum_{n=1}^{40}\left(P_n\right)^{\frac1n} = \dfrac{3^{\alpha} - 1}{3^{\beta}}$ and $\gcd(\alpha, \beta) = 1$ , then $\alpha + \beta$ is equal to
1. (1)73
2. (2)74
3. (3)75
4. (4)76
MathsJEE Main 24 Jan 2026, Shift 1·Q15·MCQ·Moderate
Open
Arithmetic progression sum
Consider an A.P.: $a_1, a_2, \ldots, a_n$ ; $a_1 > 0$ . If $a_2 - a_1 = \dfrac{-3}{4}$ , $a_n = \dfrac14 a_1$ , and $\displaystyle\sum_{i=1}^{n} a_i = \dfrac{525}{2}$ , then $\displaystyle\sum_{i=1}^{17} a_i$ is equal to
1. (1)476
2. (2)952
3. (3)238
4. (4)136
MathsJEE Main 24 Jan 2026, Shift 2·Q5·MCQ·Moderate
Open
Arithmetic progression
Let $\alpha_1, \alpha_2, \alpha_3, \alpha_4$ be an A.P. of four terms such that each term of the A.P. and its common difference $l$ are integers. If $\alpha_1 + \alpha_2 + \alpha_3 + \alpha_4 = 48$ and $\alpha_1\alpha_2\alpha_3\alpha_4 + l^4 = 361$ then the largest term of the A.P. is equal to
1. (1)27
2. (2)24
3. (3)21
4. (4)23
We corrected a printing error in the original paper
Source printed “ $\alpha_1, \alpha_2, \alpha_3, \alpha_4 + l^4 = 361$ ” (commas instead of the product $\alpha_1\alpha_2\alpha_3\alpha_4$ ).
MathsJEE Main 24 Jan 2026, Shift 2·Q7·MCQ·Moderate
Open
Infinite geometric series
$\left(\frac13 + \frac47\right) + \left(\frac{1}{3^2} + \frac13\times\frac47 + \frac{4^2}{7^2}\right) + \left(\frac{1}{3^3} + \frac{1}{3^2}\times\frac47 + \frac13\times\frac{4^2}{7^2} + \frac{4^3}{7^3}\right) + \ldots$ upto infinite terms is equal to
1. (1) $\dfrac52$
2. (2) $\dfrac74$
3. (3) $\dfrac43$
4. (4) $\dfrac65$
MathsJEE Main 28 Jan 2026, Shift 1·Q2·MCQ·Moderate
Open
Exponential series sum
The value of $\displaystyle\sum_{k=1}^{\infty}(-1)^{k+1}\left(\frac{k(k+1)}{k!}\right)$ is:
1. (1) $2/e$
2. (2) $1/e$
3. (3) $\sqrt{e}$
4. (4) $e/2$
MathsJEE Main 28 Jan 2026, Shift 1·Q17·MCQ·Easy–moderate
Open
Arithmetic progression
The common difference of the A.P.: $a_1, a_2, \ldots, a_m$ is 13 more than the common difference of the A.P.: $b_1, b_2, \ldots, b_n$ . If $b_{31} = -277$ , $b_{43} = -385$ and $a_{78} = 327$ , then $a_1$ is equal to
1. (1)21
2. (2)24
3. (3)19
4. (4)16
MathsJEE Main 28 Jan 2026, Shift 1·Q21·Numerical·Moderate
Open
Geometric progression
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 .
MathsJEE Main 28 Jan 2026, Shift 2·Q20·MCQ·Easy–moderate
Open
Sum of geometric progression
$\dfrac{6}{3^{26}} + \dfrac{10\cdot1}{3^{25}} + \dfrac{10\cdot2}{3^{24}} + \dfrac{10\cdot2^2}{3^{23}} + \cdots + \dfrac{10\cdot2^{24}}{3}$ is equal to
1. (1) $2^{25}$
2. (2) $2^{26}$
3. (3) $3^{25}$
4. (4) $3^{26}$
MathsJEE Main 28 Jan 2026, Shift 2·Q21·Numerical·Moderate
Open
Telescoping series sum
If $\displaystyle\sum_{r=1}^{25}\left(\frac{r}{r^4 + r^2 + 1}\right) = \frac{p}{q}$ , where $p$ and $q$ are positive integers such that $\gcd(p, q) = 1$ , then $p + q$ is equal to .
MathsJEE Main 2 Apr 2026, Shift 1·Q5·MCQ·Moderate
Open
Common terms of two arithmetic progressions
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 $A \cap B$ , which are divisible by 3, is:
1. (1)4
2. (2)5
3. (3)6
4. (4)7
We corrected a printing error in the original paper
Source printed “the number of elements is $A \cap B$ ”.
MathsJEE Main 2 Apr 2026, Shift 1·Q22·Numerical·Easy–moderate
Open
General term from partial sums
If $\displaystyle\sum_{k=1}^{n} a_k = 6n^3$ , then $\displaystyle\sum_{k=1}^{6}\left(\frac{a_{k+1} - a_k}{36}\right)^2$ is equal to .
MathsJEE Main 2 Apr 2026, Shift 2·Q4·MCQ·Easy–moderate
Open
Arithmetic and geometric progressions
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:
1. (1)81
2. (2)76
3. (3)62
4. (4)55
MathsJEE Main 2 Apr 2026, Shift 2·Q5·MCQ·Moderate
Open
Sum of special series
The sum $\dfrac{1^3}{1} + \dfrac{1^3 + 2^3}{1 + 3} + \dfrac{1^3 + 2^3 + 3^3}{1 + 3 + 5} + \ldots$ up to 8 terms, is:
1. (1)70
2. (2)71
3. (3)72
4. (4)73
MathsJEE Main 4 Apr 2026, Shift 1·Q7·MCQ·Moderate
Open
Arithmetic progression sum
The first term of an A.P. of 30 non-negative terms is $\dfrac{10}{3}$ . If the sum of this A.P. is the cube of its last term, then its common difference is:
1. (1) $\dfrac{5}{87}$
2. (2) $\dfrac{25}{83}$
3. (3) $\dfrac{15}{29}$
4. (4) $\dfrac{5}{29}$
MathsJEE Main 4 Apr 2026, Shift 2·Q7·MCQ·Easy–moderate
Open
Infinite geometric series
Let $\alpha = \dfrac14 + \dfrac18 + \dfrac1{16} + \ldots\infty$ and $\beta = \dfrac13 + \dfrac19 + \dfrac1{27} + \ldots\infty$ . Then the value of $(0.2)^{\log_{\sqrt5}(\alpha)} + (0.04)^{\log_5(\beta)}$ is equal to:
1. (1)4
2. (2)5
3. (3)8
4. (4)25
MathsJEE Main 5 Apr 2026, Shift 1·Q2·MCQ·Easy–moderate
Open
Arithmetic progression sum of squares
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:
1. (1)10220
2. (2)12860
3. (3)15220
4. (4)19780
MathsJEE Main 5 Apr 2026, Shift 1·Q5·MCQ·Easy–moderate
Open
Telescoping series sum
$\displaystyle\sum_{n=1}^{10}\left(\frac{528}{n(n+1)(n+2)}\right)$ is equal to:
1. (1)65
2. (2)130
3. (3)220
4. (4)440
MathsJEE Main 5 Apr 2026, Shift 2·Q5·MCQ·Moderate
Open
Telescoping series sum
If the sum of the first 10 terms of the series $\frac{1}{1 + 1^4\times4} + \frac{2}{1 + 2^4\times4} + \frac{3}{1 + 3^4\times4} + \frac{4}{1 + 4^4\times4} + \ldots$ is $\dfrac{m}{n}$ , $\gcd(m, n) = 1$ , then $m + n$ is equal to:
1. (1)256
2. (2)264
3. (3)276
4. (4)284
MathsJEE Main 5 Apr 2026, Shift 2·Q6·MCQ·Easy–moderate
Open
Arithmetic means
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:
1. (1)129
2. (2)136
3. (3)131.50
4. (4)134
MathsJEE Main 6 Apr 2026, Shift 1·Q5·MCQ·Easy–moderate
Open
Alternating sum of cubes
The value of $1^3 - 2^3 + 3^3 - \ldots + 15^3$ is:
1. (1)1706
2. (2)1856
3. (3)1982
4. (4)2403
MathsJEE Main 6 Apr 2026, Shift 1·Q6·MCQ·Moderate
Open
Arithmetic and geometric progressions
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:
1. (1) $\dfrac{34}{9}$
2. (2) $\dfrac{34}{13}$
3. (3) $\dfrac{32}{9}$
4. (4) $\dfrac{32}{13}$
MathsJEE Main 6 Apr 2026, Shift 2·Q2·MCQ·Moderate
Open
Sum of geometric progression
Consider the quadratic equation $(n^2 - 2n + 2)x^2 - 3x + (n^2 - 2n + 2)^2 = 0$ , $n \in \mathbb{R}$ . Let $\alpha$ be the minimum value of the product of its roots and $\beta$ 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 $\alpha$ and the common ratio is $\dfrac{\alpha}{\beta}$ , is:
1. (1) $\dfrac{61}{37}$
2. (2) $\dfrac{121}{81}$
3. (3) $\dfrac{364}{243}$
4. (4) $\dfrac{1093}{729}$
MathsJEE Main 6 Apr 2026, Shift 2·Q6·MCQ·Easy–moderate
Open
Sum of special series
The sum $1 + \dfrac12\left(1^2 + 2^2\right) + \dfrac13\left(1^2 + 2^2 + 3^2\right) + \ldots$ upto 10 terms is equal to:
1. (1)130
2. (2)155
3. (3) $\dfrac{315}{2}$
4. (4) $\dfrac{325}{2}$
MathsJEE Main 8 Apr 2026, Shift 2·Q5·MCQ·Moderate
Open
Sum of arithmetic progressions
Let $\alpha = 3 + 4 + 8 + 9 + 13 + 14 + \ldots$ upto 40 terms. If $(\tan\beta)^{\frac{\alpha}{1020}}$ is a root of the equation $x^2 + x - 2 = 0$ , $\beta \in \left(0, \dfrac{\pi}{2}\right)$ , then $\sin^2\beta + 3\cos^2\beta$ is equal to:
1. (1)2
2. (2) $\dfrac74$
3. (3) $\dfrac52$
4. (4) $\dfrac32$