Sequences and Series — JEE Main PYQs

If $\frac{6}{3^{12}}+\frac{10}{3^{11}}+\frac{20}{3^{10}}+\frac{40}{3^9}+\dots ..+\frac{10240}{3}=2^n\cdot m$, where m is odd, then $\mathrm{m}.\mathrm{n}$ is equal to

Suppose $a_1,a_2,\dots ,a_{\mathrm{n}},\dots$ 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 …

Let a, b be two non-zero real numbers. If p and r are the roots of the equation $x^2-8ax+2a=0$ and q and s are the roots of the equation $x^2+12bx+6b=0$, such that $\frac{1}{p},\frac{1}{q},\frac{1}{r},\frac{1}{s}$ are in A.P., then $a^{-1}-b^{-1}$ is equal to ____.

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 ______.

If ${\sum }_{k=1}^{10}\frac{k}{k^4+k^2+1}=\frac{m}{n}$, where $m$ and $n$ are coprime, then $m+n$ is equal to

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 ______.

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 $10\mathrm{ar},{\mathrm{n}}^{\text{th }}$ term is …

$\frac{2^3-1^3}{1\times 7}+\frac{4^3-3^3+2^3-1^3}{2\times 11}+\frac{6^3-5^3+4^3-3^3+2^3-1^3}{3\times 15}+\dots +\frac{30^3-29^3+28^3-27^3+\dots +2^3-1^3}{15\times 63}$ is equal to

The sum ${\sum }_{n=1}^{21}\frac{3}{(4n-1)(4n+3)}$ is equal to

Consider two G.Ps. $2,2^2,2^3,\dots$ and $4,4^2,4^3,\dots$. of 60 and n terms respectively. If the geometric mean of all the $60+n$ terms is (2) ${}^{\frac{225}{8}}$, then ${\sum }_{k=1}^{n}k(n-k)$ is equal to :

The series of positive multiples of 3 is divided into sets : $\{3\},\{6,9,12\},\{15,18,21,24,27\},\dots$ Then the sum of the elements in the $11^{\text{th }}$ set is equal to ____ ,

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 $\mathrm{a}+\mathrm{n}=33$, then the value of n is

Let for $\mathrm{n}=1,2,\dots ,50,{\mathrm{S}}_{\mathrm{n}}$ be the sum of the infinite geometric progression whose first term is ${\mathrm{n}}^2$ and whose common ratio is $\frac{1}{(\mathrm{n}+1{)}^2}$. Then the value of …

Let $S=2+\frac{6}{7}+\frac{12}{7^2}+\frac{20}{7^3}+\frac{30}{7^4}+\dots$. then $4S$ is equal to

If $a_1,a_2,a_3\dots$ and $b_1,b_2,b_3\dots$ 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

Let ${\mathrm{A}}_1,{\mathrm{A}}_2,{\mathrm{A}}_3,\dots$ be an increasing geometric progression of positive real numbers. If $A_1{A}_3{A}_5{A}_7=\frac{1}{1296}$ and $A_2+A_4=\frac{7}{36}$, then, the value of ${\mathrm{A}}_6+{\mathrm{A}}_8+{\mathrm{A}}_{10}$ is equal …

Let $A=\{1,a_1,a_2,\dots ,a_{18},77\}$ be a set of integers with $1<a_1<a_2<\dots <a_{18}<77$. Let the set $A+A=\{x+y:x,y\in A\}$ contain exactly 39 elements. Then, the value of $a_1+a_2+\dots +a_{18}$ is equal to ____.

The sum of the infinite series $1+\frac{5}{6}+\frac{12}{6^2}+\frac{22}{6^3}+\frac{35}{6^4}+\frac{51}{6^5}+\frac{70}{6^6}+\dots$ is equal to:

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

If $\mathrm{A}={\sum }_{\mathrm{n}=1}^{\infty }\frac{1}{{\left(3+(-1{)}^{\mathrm{n}}\right)}^{\mathrm{n}}}$ and $\mathrm{B}={\sum }_{\mathrm{n}=1}^{\infty }\frac{(-1{)}^{\mathrm{n}}}{{\left(3+(-1{)}^{\mathrm{n}}\right)}^{\mathrm{n}}}$, then $\frac{\mathrm{A}}{\mathrm{B}}$ is equal to :

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 ____.

Let ${\left\{a_n\right\}}_{n=0}^{\infty }$ be a sequence such that ${\mathrm{a}}_0={\mathrm{a}}_1=0$ 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

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 ____.

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 $|a|<1$, $|b|<1$, $|c|<1$, $abc\ne 0$, then

Let $f:R\to R$ be a function defined $f(x)=\frac{2e^{2x}}{e^{2x}+e}$.

Then $f\left(\frac{1}{100}\right)+f\left(\frac{2}{100}\right)+f\left(\frac{3}{100}\right)+\dots +f\left(\frac{99}{100}\right)$ is equal to ____.

If the sum of the first ten terms of the series $\frac{1}{5}+\frac{2}{65}+\frac{3}{325}+\frac{4}{1025}+\frac{5}{2501}+\dots$ is $\frac{m}{n}$, where m and n are co-prime numbers, then $m+n$ is equal to ______.

If $\frac{1}{(20-a)(40-a)}+\frac{1}{(40-a)(60-a)}+\dots \dots +\frac{1}{(180-a)(200-a)}=\frac{1}{256}$, then the maximum value of $a$ is :

Let $a_1,a_2,a_3,\dots$ be an A.P. If ${\sum }_{r=1}^{\infty }\frac{a_r}{2^r}=4$, then $4a_2$ is equal to ____ .

If $\frac{1}{2\times 3\times 4}+\frac{1}{3\times 4\times 5}+\frac{1}{4\times 5\times 6}+\dots +\frac{1}{100\times 101\times 102}=\frac{\mathrm{k}}{101}$, then 34 k is equal to ______.

If ${\left\{{\mathrm{a}}_{\mathrm{i}}\right\}}_{\mathrm{i}=1}^{\mathrm{n}}$ where n is an even integer, is an arithmetic progression with common difference 1, and …

Let $x,y>0$. If $x^3{y}^2=2^{15}$, then the least value of $3x+2y$ is

The sum $1+2\cdot 3+3\cdot 3^2+\dots ..+10\cdot 3^9$ is equal to

If $\frac{1}{2\cdot 3^{10}}+\frac{1}{2^2\cdot 3^9}+\dots \frac{1}{2^{10}\cdot 3}=\frac{\mathrm{K}}{2^{10}\cdot 3^{10}}$, then the remainder when K is divided by 6 is

For a natural number n, let ${\mathrm{a}}_{\mathrm{n}}=19^{\mathrm{n}}-12^{\mathrm{n}}$. Then, the value of $\frac{31{\alpha }_9-{\alpha }_{10}}{57{\alpha }_8}$ is

The greatest integer less than or equal to the sum of first 100 terms of the sequence $\frac{1}{3},\frac{5}{9},\frac{19}{27},\frac{65}{81},\dots \dots$ is equal to

The sum of the series $\frac{1}{x+1}+\frac{2}{x^2+1}+\frac{2^2}{x^4+1}+\dots .+\frac{2^{100}}{x^{2^{100}}+1}$ when $x=2$ is:

If the sum of an infinite GP $\mathrm{a},\mathrm{ar},{\mathrm{ar}}^2,{\mathrm{ar}}^3,\dots \dots$ is 15 and the sum of the squares of its each term is 150, then the sum of $a{r}^2,a{r}^4,a{r}^6,\dots$. is:

If $e^{\left({\cos }^{2}x+{\cos }^{4}x+{\cos }^{6}x+\dots .\infty \right){\log }_{e}2}$ satisfies the equation $t^2-9t+8=0$, then the value of $\frac{2\sin x}{\sin x+\sqrt{3}\cos x}\left(0<x<\frac{\pi }{2}\right)$ is

Let $a_1,a_2,\dots ,a_{10}$ be an AP with common difference -3 and $b_1,b_2,\dots ,b_{10}$ be a GP with common ratio 2. Let $c_k=a_k+b_k$, $k=1,2,\dots ,10$. If $c_2=12$ and $c_3=13$, then ${\sum }_{k=1}^{10}{c}_k$ is equal to ____.

Let $\mathrm{a},\mathrm{b},\mathrm{c}$ be in arithmetic progression. Let the centroid of the triangle with vertices $(\mathrm{a},\mathrm{c}),(2,\mathrm{b})$ and $(a,b)$ be $\left(\frac{10}{3},\frac{7}{3}\right)$. If $\alpha ,\beta$ are the roots of the equation …

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 ____

If sum of the first 21 terms of the series ${\log }_{9^{1/2}}\mathrm{X}+{\log }_{9^{1/3}}\mathrm{X}+{\log }_{9^{1/4}}\mathrm{X}+\dots$, where $\mathrm{x}>0$ is 504 , then $x$ is equal to :

Let ${\left\{a_n\right\}}_{n=1}^{\infty }$ 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........

If $0<x<1$ and $y=\frac{1}{2}{x}^2+\frac{2}{3}{x}^3+\frac{3}{4}{x}^4+\dots$, then the value of $e^{1+y}$ at $x=\frac{1}{2}$ is:

If $0<\theta ,\varphi <\frac{\pi }{2},x={\sum }_{n=0}^{\infty }{\cos }^{2n}\theta ,y={\sum }_{n=0}^{\infty }{\sin }^{2n}\varphi$ and $z={\sum }_{n=0}^{\infty }{\cos }^{2n}\theta \cdot {\sin }^{2n}\varphi$ then :

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 smallest value of $n$ for which area of $A_n$ is less than one, is ____

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 :

The sum of the infinite series $1+\frac{2}{3}+\frac{7}{3^2}+\frac{12}{3^3}+\frac{17}{3^4}+\frac{22}{3^5}+\dots \dots$ is equal to :

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

If for $x,y\in R,x<0,y={\log }_{10}x+{\log }_{10}{x}^{1/3}+{\log }_{10}{x}^{1/9}+\dots$. upto $\infty$ terms and $\frac{2+4+6+\dots +2y}{3+6+9+\dots +3y}=\frac{4}{{\log }_{10}x}$, then the ordered pair $(x,y)$ is equal to