Sequences and Series — JEE Main PYQs

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 :

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 :

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:

Let $\alpha = \dfrac{1}{4} + \dfrac{1}{8} + \dfrac{1}{16} + \ldots\infty$ and $\beta = \dfrac{1}{3} + \dfrac{1}{9} + \dfrac{1}{27} + \ldots\infty$. Then the value of $(0.2)^{\log_{\sqrt{5}}(\alpha)} + (0.04)^{\log_5(\beta)}$ is equal to:

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:

$\displaystyle\sum_{n=1}^{10} \left( \frac{528}{n(n+1)(n+2)} \right)$ is equal to:

If the sum of the first 10 terms of the series $\dfrac{1}{1 + 1^4 \times 4} + \dfrac{2}{1 + 2^4 \times 4} + \dfrac{3}{1 + 3^4 \times 4} + \dfrac{4}{1 + 4^4 \times 4} + \ldots$ is $\dfrac{m}{n}$, $\gcd(m, n) = 1$, then $m + n$ is equal to :

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 :

The value of $1^3-2^3+3^3-\ldots+15^3$ is:

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:

For the functions $f(\theta)=\alpha\tan^2\theta+\beta\cot^2\theta$, and $g(\theta)=\alpha\sin^2\theta+\beta\cos^2\theta$, $\alpha>\beta>0$, let $\min\limits_{0<\theta<\frac{\pi}{2}} f(\theta)=\max\limits_{0<\theta<\pi} g(\theta)$. 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 ________.

The sum $1 + \frac{1}{2}\left(1^2 + 2^2\right) + \frac{1}{3}\left(1^2 + 2^2 + 3^2\right) + \ldots$ upto 10 terms is equal to :

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 :

Let $f$ be a polynomial function such that

$$\log_2(f(x)) = \left(\log_2\left(2 + \frac{2}{3} + \frac{2}{9} + \ldots \infty\right)\right) \cdot \log_3\left(1 + \frac{f(x)}{f\left(1/x\right)}\right),\; x > 0 \text{ and } f(6) = 37.$$

Then $\sum_{n=1}^{10} f(n)$ is equal to __________.

Let $\mathrm{a_1}, \mathrm{a_2}, \mathrm{a_3}, \ldots$ be in an A.P. such that $\sum_{\mathrm{k}=1}^{12} \mathrm{a_{2k-1}} = -\frac{72}{5}\mathrm{a_1}$, $\mathrm{a_1} \neq 0$. If $\sum_{\mathrm{k}=1}^{\mathrm{n}} \mathrm{a_k} = 0$, then n is :

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 $\frac{21}{2}$. Then the number of terms which are integers in the A.P. is :

If the sum of the first 10 terms of the series $\frac{4 \cdot 1}{1+4 \cdot 1^4}+\frac{4 \cdot 2}{1+4 \cdot 2^4}+\frac{4 \cdot 3}{1+4 \cdot 3^4}+\ldots$ is $\frac{\mathrm{m}}{\mathrm{n}}$, where $\gcd(\mathrm{m}, \mathrm{n})=1$, then $\mathrm{m}+\mathrm{n}$ is equal to _________.

Let $a_1, a_2, a_3, \ldots$ be a G.P. of increasing positive numbers. If $a_3 a_5 = 729$ and $a_2 + a_4 = \frac{111}{4}$, then $24\,(a_1 + a_2 + a_3)$ is equal to

The sum $1 + 3 + 11 + 25 + 45 + 71 + \ldots$ upto 20 terms, is equal to

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

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

$1 + 3 + 5^2 + 7 + 9^2 + \ldots$ upto 40 terms is equal to

If the sum of the first 20 terms of the series $$\frac{4\cdot 1}{4+3\cdot 1^2+1^4}+\frac{4\cdot 2}{4+3\cdot 2^2+2^4}+\frac{4\cdot 3}{4+3\cdot 3^2+3^4}+\frac{4\cdot 4}{4+3\cdot 4^2+4^4}+\ldots$$ is $\dfrac{m}{n}$, where $m$ and $n$ are coprime, then $m+n$ is equal to :

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 AP's 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

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 $\frac{1}{24}(x_1\, x_2\, x_3\, x_4)$ is:

Let $a_n$ be the $n^{\text{th}}$ term of an A.P. If $S_n = a_1 + a_2 + a_3 + \ldots + a_n = 700$, $a_6 = 7$ and $S_7 = 7$, then $a_n$ is equal to :

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 :

If $\frac{1}{1^4} + \frac{1}{2^4} + \frac{1}{3^4} + \ldots \infty = \frac{\pi^4}{90}$,

$\frac{1}{1^4} + \frac{1}{3^4} + \frac{1}{5^4} + \ldots \infty = \alpha$,

$\frac{1}{2^4} + \frac{1}{4^4} + \frac{1}{6^4} + \ldots \infty = \beta$,

then $\frac{\alpha}{\beta}$ is equal to

If $\sum_{r=1}^{n} T_r=\frac{(2n-1)(2n+1)(2n+3)(2n+5)}{64}$, then $\lim_{n \to \infty} \sum_{r=1}^{n}\left(\frac{1}{T_r}\right)$ is equal to :

Let $a_1, a_2, a_3, \ldots$ be a G.P. of increasing positive terms. If $a_1 a_5=28$ and $a_2+a_4=29$, then $a_6$ is equal to :

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 :

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

The roots of the quadratic equation $3x^2 - \mathrm{p}x + \mathrm{q} = 0$ 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 $\mathrm{q} - 2\mathrm{p}$ is equal to _________.

Let $S_n = \frac{1}{2}+\frac{1}{6}+\frac{1}{12}+\frac{1}{20}+\ldots$ 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 $20^{\text{th}}$ and $15^{\text{th}}$ terms of the A.P. is

If $7 = 5 + \dfrac{1}{7}(5+\alpha) + \dfrac{1}{7^2}(5 + 2\alpha) + \dfrac{1}{7^3}(5 + 3\alpha) + \ldots\ldots \infty$, then the value of $\alpha$ is :

In an arithmetic progression, if $\mathrm{S_{40}} = 1030$ and $\mathrm{S_{12}} = 57$, then $\mathrm{S_{30}} - \mathrm{S_{10}}$ is equal to :

Let $<a_n>$ 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

Let $T_r$ be the $r^{\text{th}}$ 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

For positive integers $n$, if $4a_n = (n^2 + 5n + 6)$ and $S_n = \sum_{k=1}^{n} \left(\frac{1}{a_k}\right)$, then the value of $507\, S_{2025}$ is :

The interior angles of a polygon with $n$ sides, are in an A.P. with common difference $6°$. If the largest interior angle of the polygon is $219°$, then $n$ is equal to __________.

Let the first three terms 2, $p$ and $q$, with $q \neq 2$, of a G.P. be respectively the $7^{\text{th}}$, $8^{\text{th}}$ and $13^{\text{th}}$ terms of an A.P. If the $5^{\text{th}}$ term of the G.P. is the $n^{\text{th}}$ term of the A.P., then $n$ is equal to :

Let $a=1+\frac{^{2}C_{2}}{3!}+\frac{^{3}C_{2}}{4!}+\frac{^{4}C_{2}}{5!}+\ldots$,

$b=1+\frac{^{1}C_{0}+{}^{1}C_{1}}{1!}+\frac{^{2}C_{0}+{}^{2}C_{1}+{}^{2}C_{2}}{2!}+\frac{^{3}C_{0}+{}^{3}C_{1}+{}^{3}C_{2}+{}^{3}C_{3}}{3!}+\ldots$

Then $\frac{2b}{a^{2}}$ is equal to ________.

The value of $\frac{1 \times 2^2 + 2 \times 3^2 + \ldots + 100 \times (101)^2}{1^2 \times 2 + 2^2 \times 3 + \ldots + 100^2 \times 101}$ is

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

If $\frac{1}{\sqrt{1}+\sqrt{2}} + \frac{1}{\sqrt{2}+\sqrt{3}} + \ldots + \frac{1}{\sqrt{99}+\sqrt{100}} = \mathrm{m}$ and $\frac{1}{1 \cdot 2} + \frac{1}{2 \cdot 3} + \ldots + \frac{1}{99 \cdot 100} = \mathrm{n}$, then the point $(\mathrm{m}, \mathrm{n})$ lies on the line

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 ________.

For $x \geq 0$, 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 :

If $1 + \dfrac{\sqrt{3} - \sqrt{2}}{2\sqrt{3}} + \dfrac{5 - 2\sqrt{6}}{18} + \dfrac{9\sqrt{3} - 11\sqrt{2}}{36\sqrt{3}} + \dfrac{49 - 20\sqrt{6}}{180} + \ldots \text{upto } \infty = 2 + \left(\sqrt{\dfrac{b}{a}} + 1\right)\log_{\mathrm{e}}\left(\dfrac{a}{b}\right)$, where $a$ and $b$ are integers with $\gcd(a, b) = 1$, then $11a + 18b$ is equal to ________.

Let the first term of a series be $T_1 = 6$ and its $r^{\text{th}}$ term $T_r = 3T_{r-1} + 6^r$, $r = 2, 3, ......, n$. If the sum of the first $n$ terms of this series is $\dfrac{1}{5}\left(n^2 - 12n + 39\right)\left(4 \cdot 6^n - 5 \cdot 3^n + 1\right)$, then $n$ is equal to ________.

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 :

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 :

If $S(x)=(1+x)+2(1+x)^2+3(1+x)^3+\cdots+60(1+x)^{60}$, $x \ne 0$, and $(60)^2 S(60)=\mathrm{a(b)^b+b}$, where $\mathrm{a, b} \in \mathbf{N}$, then $(\mathrm{a+b})$ equal to ________.

If the range of $f(\theta)=\frac{\sin^4\theta+3\cos^2\theta}{\sin^4\theta+\cos^2\theta}$, $\theta\in\mathbb{R}$ is $[\alpha, \beta]$, then the sum of the infinite G.P., whose first term is 64 and the common ratio is $\frac{\alpha}{\beta}$, is equal to __________.

Let the positive integers be written in the form :

[Figure: triangular arrangement of positive integers — row 1: 1; row 2: 2, 3; row 3: 4, 5, 6; row 4: 7, 8, 9, 10; and so on]

If the $k^{\text{th}}$ row contains exactly $k$ numbers for every natural number $k$, then the row in which the number 5310 will be, is __________.

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 $4^{\text{th}}$, $6^{\text{th}}$ and $8^{\text{th}}$ terms is equal to :

An arithmetic progression is written in the following way

$$\begin{array}{ccccccc} & & & 2 & & & \\ & & 5 & & 8 & & \\ & 11 & & 14 & & 17 & \\ 20 & & 23 & & 26 & & 29 \end{array}$$ $$- - - - - - - - - - - - - -$$ $$- - - - - - - - - - - - - - - -$$

The sum of all the terms of the $10^{\text{th}}$ row is ________.

If the sum of the series $\frac{1}{1\cdot(1+\mathrm{d})}+\frac{1}{(1+\mathrm{d})(1+2\mathrm{d})}+\ldots+\frac{1}{(1+9\mathrm{d})(1+10\mathrm{d})}$ is equal to 5, then $50\mathrm{d}$ is equal to :

If a function $f$ satisfies $f(\mathrm{m}+\mathrm{n})=f(\mathrm{m})+f(\mathrm{n})$ for all $\mathrm{m}, \mathrm{n}\in\mathbf{N}$ and $f(1)=1$, then the largest natural number $\lambda$ such that $\sum\limits_{\mathrm{k}=1}^{2022} f(\lambda+\mathrm{k})\le(2022)^2$ is equal to ________.

Let $a, ar, ar^2, \ldots\ldots$ be an infinite G.P. If $\sum\limits_{n=0}^{\infty} ar^n = 57$ and $\sum\limits_{n=0}^{\infty} a^3r^{3n} = 9747$, then $a+18r$ is equal to

If $\left(\dfrac{1}{\alpha+1}+\dfrac{1}{\alpha+2}+\ldots\ldots+\dfrac{1}{\alpha+1012}\right)-\left(\dfrac{1}{2\cdot 1}+\dfrac{1}{4\cdot 3}+\dfrac{1}{6\cdot 5}+\ldots\ldots+\dfrac{1}{2024\cdot 2023}\right)=\dfrac{1}{2024}$, then $\alpha$ is equal to __________.

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 $a, b$ and $c$ is :

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 ________.

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 :

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 ________.

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 :

If $8 = 3 + \dfrac{1}{4}(3 + p) + \dfrac{1}{4^2}(3 + 2p) + \dfrac{1}{4^3}(3 + 3p) + \cdots \infty$, then the value of $p$ is ________.

In an A.P., the sixth term $a_6 = 2$. If the product $a_1 a_4 a_5$ is the greatest, then the common difference of the A.P. is equal to

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

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 :

Let $\alpha=1^2+4^2+8^2+13^2+19^2+26^2+...$ upto 10 terms and $\beta=\sum\limits_{n=1}^{10} n^4$. If $4\alpha-\beta=55k+40$, then $k$ is equal to ________.

Let $a$ and $b$ be 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

Let $S_n$ be the sum to $n$-terms of an arithmetic progression $3, 7, 11, \ldots$ If $40<\left(\frac{6}{n(n+1)}\sum_{k=1}^{n} S_k\right)<42$, then $n$ equals ______.

The sum of the series $\frac{1}{1-3 \cdot 1^2+1^4}+\frac{2}{1-3 \cdot 2^2+2^4}+\frac{3}{1-3 \cdot 3^2+3^4}+\ldots$ up to 10-terms is

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