Sequences and Series — JEE Main PYQs

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

If $\gcd (\mathrm{m},\mathrm{n})=1$ and $1^2-2^2+3^2-4^2+\dots +(2021{)}^2-(2022{)}^2+(2023{)}^2=1012{\mathrm{m}}^2\mathrm{n}$, then ${\mathrm{m}}^2-{\mathrm{n}}^2$ is equal to

Let ${\mathrm{S}}_{\mathrm{k}}=\frac{1+2+\dots ..+\mathrm{K}}{\mathrm{K}}$ and
${\sum }_{\mathrm{j}=1}^{\mathrm{n}}{\mathrm{S}}_{\mathrm{j}}^2=\frac{\mathrm{n}}{\mathrm{A}}\left({\mathrm{Bn}}^2+\mathrm{Cn}+\mathrm{D}\right)$, where …

The sum of the first 20 terms of the series $5+11+19+29+41+\dots$ is

Let ${\mathrm{a}}_{\mathrm{n}}$ be the ${\mathrm{n}}^{\text{th}}$ term of the series $5+8+14+23+35+50+\dots$ and ${\mathrm{S}}_{\mathrm{n}}={\sum }_{\mathrm{k}=1}^{\mathrm{n}}{\mathrm{a}}_{\mathrm{k}}$. Then ${\mathrm{S}}_{30}-{\mathrm{a}}_{40}$ is equal to

Let $0<\mathrm{z}<\mathrm{y}<\mathrm{x}$ be three real numbers such that $\frac{1}{\mathrm{x}},\frac{1}{\mathrm{y}},\frac{1}{\mathrm{z}}$ are in an arithmetic progression and $x$, $\sqrt{2}y$, $z$ are in a geometric progression. If $xy+yz+zx=\frac{3}{\sqrt{2}}xyz$, then …

If ${\mathrm{S}}_{\mathrm{n}}=4+11+21+34+50+\dots$. to n terms, then $\frac{1}{60}\left({\mathrm{S}}_{29}-{\mathrm{S}}_9\right)$ is equal to

Suppose ${\mathrm{a}}_1,{\mathrm{a}}_2,2,{\mathrm{a}}_3,{\mathrm{a}}_4$ 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 $\frac{49}{2}$, t …

Let the first term $a$ and the common ratio r of a geometric progression be positive integers. If the sum of its squares of first three terms is 33033, then the sum of these three terms is equal to

The sum of all those terms, of the arithmetic progression $3,8,13,\dots ,373$, which are not divisible by 3, is equal to ______.

Let $⟨a_n⟩$ be a sequence such that $a_1+a_2+\dots +a_n=\frac{n^2+3n}{(n+1)(n+2)}$. If $28{\sum }_{k=1}^{10}\frac{1}{a_k}=p_1{p}_2{p}_3\dots p_m$, where $p_1,p_2,\dots ,p_m$ are the first $m$ prime numbers, then $m$ is equal to

Let the positive numbers ${\mathrm{a}}_1,{\mathrm{a}}_2,{\mathrm{a}}_3,{\mathrm{a}}_4$ and ${\mathrm{a}}_5$ be in a G.P. Let their mean and variance be $\frac{31}{10}$ and $\frac{m}{n}$ respectively, where m and n are co-prime. If the mean of their reciprocals is …

Let $a_1,a_2,a_3,\dots$ be a G.P. of increasing positive numbers. Let the sum of its $6^{\text{th}}$ and $8^{\text{th}}$ terms be 2 and the product of its $3^{\text{rd}}$ and $5^{\text{th}}$ terms be $\frac{1}{9}$. Then $6(a_2+a_4)(a_4+a_6)$ is equal to

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 ${\sum }_{i=1}^{10}{s}_i$ is equal to

The sum to 20 terms of the series
$2.2^2-3^2+2.4^2-5^2+2.6^2-$ _____ is equal to _____.

Let ${\mathrm{A}}_1$ and ${\mathrm{A}}_2$ be two arithmetic means and ${\mathrm{G}}_1,{\mathrm{G}}_2,{\mathrm{G}}_3$ be three geometric means of two distinct positive numbers. Then …

If the sum of the series …

Let a, b, c and d be positive real numbers such that $a+b+c+d=11$. If the maximum value of $a^5{b}^3{c}^{2}d$ is $3750\beta$, then the value of $\beta$ is

For $\mathrm{k}\in \mathbb{N}$, if the sum of the series
$1+\frac{4}{\mathrm{k}}+\frac{8}{{\mathrm{k}}^2}+\frac{13}{{\mathrm{k}}^3}+\frac{19}{{\mathrm{k}}^4}+\dots$ is 10, then the value of k is

Let ${\mathrm{x}}_1,{\mathrm{x}}_2,\dots ,{\mathrm{x}}_{100}$ be in an arithmetic progression, with ${\mathrm{x}}_1=2$ and their mean equal to 200. If $y_i=i(x_i-i)$, $1\le i\le 100$, then the mean of ${\mathrm{y}}_1,{\mathrm{y}}_2,\dots ,{\mathrm{y}}_{100}$ is

Let $\mathrm{S}=109+\frac{108}{5}+\frac{107}{5^2}+\dots +\frac{2}{5^{107}}+\frac{1}{5^{108}}$. Then the value of $(16\mathrm{S}-(25{)}^{-54})$ is equal to

For three positive integers $p,q,r,x^{p{q}^2}=y^{qr}=z^{p^{2}r}$ and $\mathrm{r}=\mathrm{pq}+1$ such that $3,3{\log }_{\mathrm{y}}\mathrm{x},3{\log }_{\mathrm{z}}\mathrm{y},7{\log }_{\mathrm{x}}\mathrm{z}$ are in A.P. with common difference $\frac{1}{2}$. Then …

The $4^{\text{th}}$ term of GP is 500 and its common ratio is $\frac{1}{m},m\in N$. Let $S_n$ denote the sum of the first $n$ terms of this GP. If ${\mathrm{S}}_6>{\mathrm{S}}_5+1$ and ${\mathrm{S}}_7<{\mathrm{S}}_6+\frac{1}{2}$, then the number of possible values …

If $a_n=\frac{-2}{4n^2-16n+15}$, then $a_1+a_2+\dots +a_{25}$ is equal to:

Let ${\sum }_{n=0}^{\infty }\frac{n^3((2n)!)+(2n-1)(n!)}{(n!)((2n)!)}=ae+\frac{b}{e}+c$,
where $a,b,c\in \mathbb{Z}$ and $e={\sum }_{n=0}^{\infty }\frac{1}{n!}$. Then $a^2-b+c$ is equal to ____.

The sum ${\sum }_{n=1}^{\infty }\frac{2n^2+3n+4}{(2n)!}$ is equal to :

The sum of the common terms of the following three arithmetic progressions.
$3,7,11,15,\dots ,399$,
$2,5,8,11,\dots ,359$ and
$2,7,12,17,\dots ,197$, is equal to ______.

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 $\frac{1}{f(2022)}+\frac{1}{f(2028)}$ is equal to

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 $S={\sum }_{n=1}^{10}\frac{b_n}{2^n}$ and $T={\sum }_{n=1}^8\frac{n}{2^{n-1}}$, then $2^7(2S-T)$ is equal to

Let $\{a_k\}$ and $\{b_k\}$, $k\in \mathbb{N}$, be two G.P.s with common ratio $r_1$ and $r_2$ respectively such that ${\mathrm{a}}_1={\mathrm{b}}_1=4$ and ${\mathrm{r}}_1<{\mathrm{r}}_2$. Let …

Suppose f is a function satisfying $\mathrm{f}(\mathrm{x}+\mathrm{y})=\mathrm{f}(\mathrm{x})+\mathrm{f}(\mathrm{y})$ for all $x,y\in \mathbb{N}$ and $f(1)=\frac{1}{5}$. If ${\sum }_{n=1}^m\frac{f(n)}{n(n+1)(n+2)}=\frac{1}{12}$, then $m$ is equal to ______.

Let ${\mathrm{a}}_1,{\mathrm{a}}_2,{\mathrm{a}}_3,\dots$ 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 ______.

For the two positive numbers $\mathrm{a},\mathrm{b}$, if $\mathrm{a},\mathrm{b}$ and $\frac{1}{18}$ are in a geometric progression, while $\frac{1}{\mathrm{a}},10$ and $\frac{1}{\mathrm{b}}$ are in an arithmetic progression, then, $16\mathrm{a}+12\mathrm{b}$ is …

Let $f(x)$ be a function such that $f(x+y)=f(x)\cdot f(y)$ for all $x,y\in N$. If $f(1)=3$ and ${\sum }_{k=1}^{n}f(k)=3279$, then the value of n is

If $f(x)=\frac{2^{2x}}{2^{2x}+2}$, $x\in R$, then $f\left(\frac{1}{2023}\right)+f\left(\frac{2}{2023}\right)+\dots +f\left(\frac{2022}{2023}\right)$ is equal to

If $\frac{1^3+2^3+3^3+\dots \text{upto }n\text{ terms}}{1\cdot 3+2\cdot 5+3\cdot 7+\dots \text{upto }n\text{ terms}}=\frac{9}{5}$, then the value of $n$ is

Let ${\mathrm{a}}_1,{\mathrm{a}}_2,{\mathrm{a}}_3,\dots$ be an A.P. If ${\mathrm{a}}_7=3$, the product ${\mathrm{a}}_1{\mathrm{a}}_4$ is minimum and the sum of its first n terms is zero, then $\mathrm{n}!-4{\mathrm{a}}_{\mathrm{n}(\mathrm{n}+2)}$ is equal to :

The sum
$1^2-2.3^2+3.5^2-4.7^2+5.9^2-\dots \dots \dots +15.29^2$ is ______.

The parabolas : $a{x}^2+2bx+cy=0$ and $d{x}^2+2ex+fy=0$ intersect on the line $\mathrm{y}=1$. If $\mathrm{a},\mathrm{b},\mathrm{c},\mathrm{d},\mathrm{e},\mathrm{f}$ are positive real numbers and $\mathrm{a},\mathrm{b},\mathrm{c}$ are in G.P., then

Let $\mathrm{a},\mathrm{b},\mathrm{c}>1,{\mathrm{a}}^3,{\mathrm{b}}^3$ and ${\mathrm{c}}^3$ be in A.P., and ${\log }_{\mathrm{a}}\mathrm{b}$, ${\log }_{c}a$ and ${\log }_{b}c$ be in G.P. If the sum of first 20 terms of an A.P., whose first term is …

The $8^{\text{th }}$ common term of the series
$S_1=3+7+11+15+19+\dots .$,
$S_2=1+6+11+16+21+\dots$.
is ____ .

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

Let ${\mathrm{a}}_1,{\mathrm{a}}_2,\dots \dots ,{\mathrm{a}}_{\mathrm{n}}$ be in A.P. If ${\mathrm{a}}_5=2{\mathrm{a}}_7$ and ${\mathrm{a}}_{11}=18$, then …

The sum to 10 terms of the series $\frac{1}{1+1^2+1^4}+\frac{2}{1+2^2+2^4}+\frac{3}{1+3^2+3^4}+\dots$. is:-

Let ${\mathrm{a}}_1=8,{\mathrm{a}}_2,{\mathrm{a}}_3,\dots {\mathrm{a}}_{\mathrm{n}}$ be an 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 ______.

Let ${\mathrm{A}}_1,{\mathrm{A}}_2,{\mathrm{A}}_3$ be the three A.P. with the same common difference d and having their first terms as $\mathrm{A},\mathrm{A}+1,\mathrm{A}+2$, respectively. Let $\mathrm{a},\mathrm{b},\mathrm{c}$ be the …

Let ${\left\{a_{\mathrm{n}}\right\}}_{\mathrm{n}=0}^{\infty }$ be a sequence such that $a_0=a_1=0$ and $a_{\mathrm{n}+2}=3a_{\mathrm{n}+1}-2a_{\mathrm{n}}+1,\forall \mathrm{n}\ge 0$.
Then $a_{25}{a}_{23}-2a_{25}{a}_{22}-2a_{23}{a}_{24}+4a_{22}{a}_{24}$ is equal to:

${\sum }_{\mathrm{r}=1}^{20}\left({\mathrm{r}}^2+1\right)(\mathrm{r}!)$ is equal to:

Let $\alpha ,\beta$ ($\alpha >\beta$) be the roots of the quadratic equation ${\mathrm{x}}^2-\mathrm{x}-4=0$. If ${\mathrm{P}}_{\mathrm{n}}={\alpha }^{\mathrm{n}}-{\beta }^{\mathrm{n}},\mathrm{n}\in \mathbb{N}$, then …

Consider the sequence ${\mathrm{a}}_1,{\mathrm{a}}_2,{\mathrm{a}}_3,\dots$ such that ${\mathrm{a}}_1=1,{\mathrm{a}}_2=2$ and ${\mathrm{a}}_{\mathrm{n}+2}=\frac{2}{{\mathrm{a}}_{\mathrm{n}+1}}+{\mathrm{a}}_{\mathrm{n}}$ for $\mathrm{n}=1,2,3,\dots$. If …