Sequences and Series — JEE Main PYQs

If $0<x<1$, then $\frac{3}{2}{x}^2+\frac{5}{3}{x}^3+\frac{7}{4}{x}^4+\dots$, is equal to

The sum of the series ${\sum }_{n=1}^{\infty }\frac{n^2+6n+10}{(2n+1)!}$ is equal to :

If the arithmetic mean and geometric mean of the $p^{\text{th }}$ and $q^{\text{th }}$ terms of the sequence $-16,8,-4,2,\dots$ satisfy the equation $4x^2-9x+5=0$, then $p+q$ is equal to ______

Let $\alpha$ and $\beta$ be two real numbers such that $\alpha +\beta =1$ and $\alpha \beta =-1$. Let $P_n=(\alpha {)}^n+(\beta {)}^n,P_{n-1}=11$ and $P_{n+1}=29$ for some integer $n\ge 1$. Then, the value of $p_n^2$ is ______

Let $a_1,a_2,a_3,\dots$ be an A.P. If $\frac{a_1+a_2+\dots +a_{10}}{a_1+a_2+\dots +a_p}=\frac{100}{p^2},p\ne 10$, then $\frac{a_{11}}{a_{10}}$ is equal to:

If $\mathrm{S}=\frac{7}{5}+\frac{9}{5^2}+\frac{13}{5^3}+\frac{19}{5^4}+\dots .$. , then 160 S is equal to ____.

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:

The sum of 10 terms of the series
is :

The mean of 10 numbers $7\times 8,10\times 10,13\times 12,16\times 14,\dots$ is ____.

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 ____.

The value of $4+\frac{1}{5+\frac{1}{4+\frac{1}{5+\frac{1}{4+\dots \infty }}}}$ is :

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 ;

If the value of ${\left(1+\frac{2}{3}+\frac{6}{3^2}+\frac{10}{3^3}+\dots ..\text{ upto }\infty \right)}^{{\log }_{(0.25)}\left(\frac{1}{3}+\frac{1}{3^2}+\frac{1}{3^3}+\dots .\text{ upto }\infty \right)}$ is I , then ${\mathrm{I}}^2$ is equal to

If $\tan \left(\frac{\pi }{9}\right),\mathrm{x},\tan \left(\frac{7\pi }{18}\right)$ are in arithmetic progression and $\tan \left(\frac{\pi }{9}\right),\mathrm{y},\tan \left(\frac{5\pi }{18}\right)$ are also in arithmetic progression, then $|x-2y|$ is equal to :

Let $a_1,a_2,\dots \dots ,a_{21}$ 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:

Let $S_n=1\cdot (n-1)+2\cdot (n-2)+3\cdot (n-3)+\dots +(n-1)\cdot 1,n\ge 4$. The sum ${\sum }_{n=4}^{\infty }\left(\frac{2S_n}{n!}-\frac{1}{(n-2)!}\right)$ is equal to:

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 ____

Let $S_n(x)={\log }_{a^{1/2}}x+{\log }_{a^{1/3}}x+{\log }_{a^{1/6}}x+{\log }_{a^{1/11}}x+{\log }_{a^{1/18}}x+{\log }_{a^{1/27}}x+\dots$. up to $n$-terms. Where $a>$ 1. If $S_{24}(x)=1093$ and $S_{12}(2x)=265$, then the value of $a$ is equal to.....

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 ______

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 :

If ${\sum }_{r=1}^{10}r!(r^3+6r^2+2r+5)=\alpha (11!)$, then the value of $\alpha$ is equal to.......

If $\alpha ,\beta$ are natural numbers such that $100^{\alpha }-199\beta =(100)(100)+(99)(101)+(98)(102)+\dots ..+(1)(199)$, then the slope of the line passing through $(\alpha ,\beta )$ and origin is :

$\frac{1}{3^2-1}+\frac{1}{5^2-1}+\frac{1}{7^2-1}+\dots ..+\frac{1}{(201{)}^2-1}$ is equal to :

The value of $3+\frac{1}{4+\frac{1}{3+\frac{1}{4+\frac{1}{3+\dots ..\infty }}}}$ is equal to :

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:

The greatest positive integer k , for which $49^{\mathrm{k}}+1$ is a factor of the sum $49^{125}+49^{124}+\dots \dots +49^2+49+1$ is:

If the sum of the first 40 terms of the series, $3+4+8+9+13+14+18+19+\dots$. is $(102)\mathrm{m}$, then m is equal to:

Let $a_1,a_2,a_3,\dots$ be a G.P. such that $a_1<0$, $a_1+a_2=4$ and $a_3+a_4=16$. If ${\sum }_{i=1}^9{a}_i=4\lambda$, then $\lambda$ is equal to:

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:

The sum ${\sum }_{k=1}^{20}(1+2+3+\dots .+k)$ is

If the $10^{\text{th}}$ term of an A.P. is $\frac{1}{20}$ and its $20^{\text{th}}$ term is $\frac{1}{10}$, then the sum of its first 200 terms is:

The sum ${\sum }_{n=1}^7\frac{n(n+1)(2n+1)}{4}$ is equal to

If $\mathrm{x}={\sum }_{\mathrm{n}=0}^{\infty }(-1{)}^{\mathrm{n}}{\tan }^{2\mathrm{n}}\theta$ and $\mathrm{y}={\sum }_{\mathrm{n}=0}^{\infty }{\cos }^{2\mathrm{n}}\theta$, for $0<\theta <\frac{\pi }{4}$, then:

Let ${\mathrm{a}}_{\mathrm{n}}$ be the ${\mathrm{n}}^{\text{th }}$ term of a G.P. of positive terms. If ${\sum }_{\mathrm{n}=1}^{100}{\mathrm{a}}_{2\mathrm{n}+1}=200$ and ${\sum }_{\mathrm{n}=1}^{100}{\mathrm{a}}_{2\mathrm{n}}=100$, then ${\sum }_{n=1}^{200}{a}_n$ is equal to:

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

The product $2^{\frac{1}{4}}\cdot 4^{\frac{1}{16}}\cdot 8^{\frac{1}{48}}\cdot 16^{\frac{1}{128}}\dots \dots \dots$ to $\infty$ is equal to

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:

If the sum of the first 20 terms of the series ${\log }_{(7^{1/2})}x+{\log }_{(7^{1/3})}x+{\log }_{(7^{1/4})}x+\dots$ is 460, then x is equal to:

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:

The value of $(0.16{)}^{{\log }_{2.5}\left(\frac{1}{3}+\frac{1}{3^2}+\frac{1}{3^3}+\dots \text{ to }\infty \right)}$ is equal to ______.

If the sum of the series $20+19\frac{3}{5}+19\frac{1}{5}+18\frac{4}{5}+\dots$ upto ${\mathrm{n}}^{\text{th }}$ term is 488 and the ${\mathrm{n}}^{\text{th }}$ term is negative, then:

If $m$ arithmetic means (A.Ms) and three geometric means (G.Ms) are inserted between 3 and 243 such that $4^{\text{th}}$ A.M. is equal to $2^{\text{nd}}$ G.M., then m is equal to ____.

The common difference of the A.P. $b_1,b_2,\dots ,b_m$ is 2 more than the common difference of A.P. $a_1,a_2,\dots ,a_n$. If $a_{40}=-159$, $a_{100}=-399$ and $b_{100}=a_{70}$, then $b_1$ is equal to:

Suppose that a function $f:\mathbb{R}\to \mathbb{R}$ satisfies $f(x+y)=f(x)f(y)$ for all $x,y\in \mathbb{R}$ and $f(1)=3$. If ${\sum }_{i=1}^{n}f(i)=363$, then $n$ is equal to

If $2^{10}+2^9\cdot 3^1+2^8\cdot 3^2+\dots +2\cdot 3^9+3^{10}=S-2^{11}$, then $S$ is equal to

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:

If $f(x+y)=f(x)f(y)$ and ${\sum }_{x=1}^{\infty }f(x)=2,x,y\in N$ where $N$ is the set of all natural numbers, then the value of $\frac{f(4)}{f(2)}$ is:

Let $\mathrm{a},\mathrm{b},\mathrm{c},\mathrm{d}$ and p be any non zero distinct real numbers such that $\left(a^2+b^2+c^2\right)p^2-2(ab+bc+cd)p+\left(b^2+c^2+d^2\right)=0$. Then:

If $1+\left(1-2^2\cdot 1\right)+\left(1-4^2\cdot 3\right)+\left(1-6^2\cdot 5\right)+\dots \dots ..+\left(1-20^2\cdot 19\right)=\alpha -220\beta$, then an ordered pair $(\alpha ,\beta )$ is equal to:

Let $a_1,a_2,\dots a_n$ be a given A.P. whose common difference is an integer and $S_n=a_1+a_2+\dots +a_n$. If $a_1=1,a_n=300$ and $15\le n\le 50$, then the ordered pair ( $S_{n-4},a_{n-4}$ ) is equal to: