Sequences and Series — JEE Main PYQs

The sum ${\sum }_{k=1}^{20}k\frac{1}{2^k}$ is equal to:

If three distinct numbers $a,b,c$ are in G.P. and the equations $a{x}^2+2bx+c=0$ and $d{x}^2+2ex+f=0$ have a common root, then which one of the following statements is correct?

Let the sum of the first $n$ terms of a non-constant A.P., $a_1,a_2,a_3,\dots \dots$ be $50\mathrm{n}+\frac{\mathrm{n}(\mathrm{n}-7)}{2}\mathrm{A}$, where A is a constant. If d is the common difference of this A.P., then the ordered pair ( $d,a_{50}$ ) is e …

Let ${\sum }_{k=1}^{10}f(a+k)=16\left(2^{10}-1\right)$, where the function $f$ satisfies $f(x+y)=f(x)f(y)$ for all natural numbers $x,y$ and $f(1)=2$. Then the natural number ' $a$ ' is:

The sum of all two digit positive numbers which when divided by 7 yield 2 or 5 as remainder is:

Let $\alpha$ and $\beta$ be the roots of the quadratic equation $x^2\sin \theta -x(\sin \theta \cos \theta +1)+\cos \theta =0$ $(0<\theta <45^0)$, and $\alpha <\beta$. Then ${\sum }_{n=0}^{\infty }\left({\alpha }^n+\frac{(-1{)}^n}{{\beta }^n}\right)$ is equal to:

If $19^{\text{th}}$ term of a non-zero A.P. is zero, then its $(49^{\text{th}}\text{ term}):(29^{\text{th}}\text{ term})$ is:

If $\alpha$ and $\beta$ are the roots of the equation $375x^2-25x-2=0$, then ${\lim }_{n\to \infty }{\sum }_{r=1}^n{\alpha }^r+{\lim }_{n\to \infty }{\sum }_{r=1}^n{\beta }^r$ is equal to :

Let $S_n$ denote the sum of the first $n$ terms of an A.P.. If $S_4=16$ and $S_6=-48$, then $S_{10}$ is equal to :

Let $\mathrm{a},\mathrm{b}$ and c be in G.P with common ratio r, where $\mathrm{a}\ne 0$ and $0<\mathrm{r}\le \frac{1}{2}$. If $3\mathrm{a},7\mathrm{b}$ and 15 c are the first three terms of an A.P., then the $4^{\text{th }}$ term of this A.P is

The sum $1+\frac{1^3+2^3}{1+2}+\frac{1^3+2^3+3^3}{1+2+3}+\dots +\frac{1^3+2^3+3^3+\dots +15^3}{1+2+3+\dots +15}-\frac{1}{2}(1+2+3+\dots +15)$ is equal to

The sum of an infinite geometric series with positive terms is 3 and the sum of the cubes of its terms is $\frac{27}{19}$. Then the common ratio of this series is:

Let $a_1,a_2,\dots ,a_{10}$ be a G.P. If $\frac{a_3}{a_1}=25$, then $\frac{a_9}{a_5}$ equals:

The product of three consecutive terms of a G.P. is 512. If 4 is added to each of the first and the second of these terms, the three terms now form an A.P. Then the sum of the original three terms of the given G.P. is:

Let ${\mathrm{S}}_{\mathrm{k}}=\frac{1+2+3+\dots ..+\mathrm{k}}{\mathrm{k}}$. If ${\mathrm{S}}_1^2+{\mathrm{S}}_2^2+\dots ..+{\mathrm{S}}_{10}^2=\frac{5}{12}\mathrm{A}$, then A equal to:

If $a_1,a_2,a_3,\dots$ are in A.P. such that $a_1+a_7+a_{16}=40$, then the sum of the first 15 terms of this A.P. is:

If $\alpha ,\beta$ and $\gamma$ are three consecutive terms of a non-constant G.P. such that the equations $\alpha x^2+2\beta x+\gamma =0$ and $x^2+x-1=0$ have a common root, then $\alpha (\beta +\gamma )$ is equal to :

If the sum of the first 15 terms of the series ${\left(\frac{3}{4}\right)}^3+{\left(1\frac{1}{2}\right)}^3+{\left(2\frac{1}{4}\right)}^3+3^3+{\left(3\frac{3}{4}\right)}^3+\dots .$ is equal to 225 k , then k is equal to :

Some identical balls are arranged in rows to form an equilateral triangle. The first row consists of one ball, the second row consists of two balls and so on. If 99 more identical balls are added to the total number of balls used in forming the equilateral triangle, then all thes …

If the sum and product of the first three term in an A.P. are 33 and 1155, respectively, then a value of its $11^{\text{th}}$ term is

The sum of the series $1+2\times 3+3\times 5+4\times 7+\dots$ upto $11^{\text{th}}$ term is

The sum of the following series $1+6+\frac{9\left(1^2+2^2+3^2\right)}{7}+\frac{12\left(1^2+2^2+3^2+4^2\right)}{9}+\frac{15\left(1^2+2^2+\dots .+5^2\right)}{11}+\dots$ up to 15 terms, is:

Let $\mathrm{a},\mathrm{b}$ and c be the $7^{\text{th }},11^{\text{th }}$ and $13^{\text{th }}$ terms respectively of a non - constant A.P. If these are also the three consecutive terms of a G.P. then $\frac{a}{c}$ is equal to:

If $\mathrm{a},\mathrm{b}$ and c be three distinct numbers in G.P. and $\mathrm{a}+\mathrm{b}+\mathrm{c}=\mathrm{xb}$ then x can not be

Let $a_1,a_2,\dots ,a_{30}$ be an A.P., $S={\sum }_{i=1}^{30}{a}_i$ and $T={\sum }_{i=1}^{15}{a}_{2i-1}$. If $a_5=27$ and $S-2T=75$, then $a_{10}$ is equal to

The sum

Upto $10^{\text{th }}$ term, is

If ${\mathrm{a}}_1,{\mathrm{a}}_2,{\mathrm{a}}_3,\dots ,{\mathrm{a}}_{\mathrm{n}}$ are in A.P and ${\mathrm{a}}_1+{\mathrm{a}}_4+{\mathrm{a}}_7+\dots +{\mathrm{a}}_{16}=114$, then ${\mathrm{a}}_1+{\mathrm{a}}_6+{\mathrm{a}}_{11}+{\mathrm{a}}_{16}$ is equal to

Let A be the sum of the first 20 terms and B be the sum of the first 40 terms of the series
$1^2+2\cdot 2^2+3^2+2\cdot 4^2+5^2+2\cdot 6^2+\dots .$.

If $B-2A=100\lambda$, then $\lambda$ is equal to :

Let $a_1,a_2,a_3,\dots .,a_{49}$ be in A.P. such that ${\sum }_{k=0}^{12}{a}_{4k+1}=416$ and $a_9+a_{43}=66$. If ${\mathrm{a}}_1^2+{\mathrm{a}}_2^2+\dots ..+{\mathrm{a}}_{17}^2=140\mathrm{m}$, then m is equal to :

For any three positive real numbers $\mathrm{a},\mathrm{b}$ and c ,
$9\left(25a^2+b^2\right)+25\left(c^2-3ac\right)=15b(3a+c)$.
Then :

If the $2^{\text{nd }}$, $5^{\text{th }}$ and $9^{\text{th }}$ terms of a non-constant A.P. are in G.P., then the common ratio of this G.P. is:

If the sum of the first ten terms of the series ${\left(1\frac{3}{5}\right)}^2+{\left(2\frac{2}{5}\right)}^2+{\left(3\frac{1}{5}\right)}^2+4^2+{\left(4\frac{4}{5}\right)}^2+\dots$, is $\frac{16}{5}m$, then m is equal to:

The sum of first 9 terms of the series $\frac{1^3}{1}+\frac{1^3+2^3}{1+3}+\frac{1^3+2^3+3^3}{1+3+5}+\dots \dots$. is :

Let $\alpha$ and $\beta$ be the roots of equation $x^2-6x-2=0$. If $a_n={\alpha }^n-{\beta }^n$, for $n\ge 1$, then the value of $\frac{a_{10}-2a_8}{2a_9}$ is equal to :

If m is the A.M. of two distinct real numbers $\ell$ and $\mathrm{n}(\ell ,\mathrm{n}>1)$ and ${\mathrm{G}}_1,{\mathrm{G}}_2$ and ${\mathrm{G}}_3$ are three geometric means between $\ell$ and n , then ${\mathrm{G}}_1^4+2{\mathrm{G}}_2^4+{\mathrm{G}}_3^4$ equals :

Three positive numbers from an increasing G.P. If the middle term in this G.P. is doubled, the new numbers are in A.P. Then the common ratio of the G.P. is

If $(10{)}^9+2(11{)}^1(10{)}^8+3(11{)}^2(10{)}^7+\dots +10(11{)}^9=k(10{)}^9$, then $k$ is equal to

The sum of first 20 terms of the sequence $0.7,0.77,0.777$ $\_\_\_\_$ is

Statement 1: The sum of the series $1+(1+2+4)+(4+6+9)+(9+12+16)+\dots +(361+380+400)$ is 8000.

Statement 2: ${\sum }_{k=1}^n\left(k^3-(k-1{)}^3\right)=n^3$ for any natural number n.

If 100 times the $100^{\text{th }}$ term of an AP with non zero common difference equals the 50 times its $50^{\text{th }}$ term, then the $150^{\text{th }}$ term of this AP is

A man saves Rs. 200 in each of the first three months of his service. In each of the subsequent months his saving increases by Rs. 40 more than the saving of immediately previous month. His total saving from the start of service will be Rs. 11040 after

A person is to count 4500 currency notes. Let $a_n$ denote the number of notes he counts in the $n^{\text{th}}$ minute. If $a_1=a_2=\dots =a_{10}=150$ and $a_{10},a_{11},\dots$ are in A.P. with common difference $-2$, then the time taken by him to count all notes is

The sum to the infinity of the series $1+\frac{2}{3}+\frac{6}{3^2}+\frac{10}{3^3}+\frac{14}{3^4}+\dots \dots$. is

The first two terms of a geometric progression add up to 12. The sum of the third and the fourth terms is 48. If the terms of the geometric progression are alternately positive and negative, then the first term is

In a geometric progression consisting of positive terms, each term equals the sum of the next two terms. Then the common ratio of this progression equals

The sum of the series $\frac{1}{2!}-\frac{1}{3!}+\frac{1}{4!}-\dots$ upto infinity is

Let $a_1,a_2,a_3,\dots$ be terms of an A.P. If $\frac{a_1+a_2+\cdots a_p}{a_1+a_2+\cdots +a_q}=\frac{p^2}{q^2},p\ne q$, then $\frac{a_6}{a_{21}}$ equals

If $a_1,a_2,\dots ,a_n$ are in H.P., then the expression $a_1{a}_2+a_2{a}_3+\dots +a_{n-1}{a}_n$ 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 $|a|<1,|b|<1,|c|<1$, then $x,y,z$ are in

If non-zero numbers $a,b,c$ are in H.P., then the straight line $\frac{x}{a}+\frac{y}{b}+\frac{1}{c}=0$ always passes through a fixed point. That point is