Sequences and Series — JEE Main PYQs

Let ${\mathrm{a}}_1,{\mathrm{a}}_2,{\mathrm{a}}_3,\dots$. be a G.P. of increasing positive terms such that $a_2\cdot a_3\cdot a_4=64$ and $a_1+a_3+a_5=\frac{813}{7}$.

Then ${\mathrm{a}}_3+{\mathrm{a}}_5+{\mathrm{a}}_7$ is equal to :

Let ${\mathrm{a}}_1=1$ and for $\mathrm{n}\ge 1,{\mathrm{a}}_{\mathrm{n}+1}=\frac{1}{2}{a}_n+\frac{n^2-2n-1}{n^2(n+1{)}^2}$. Then $\left|{\sum }_{n=1}^{\infty }\left(a_n-\frac{2}{n^2}\right)\right|$ is equal to

The positive integer n, for which the solutions of the equation

are two consecutive even integers, is :-

Let $a_1,\frac{a_2}{2},\frac{a_3}{2^2},\dots ,\frac{a_{10}}{2^9}$ be a G.P. of common ratio $\frac{1}{\sqrt{2}}$. If $a_1+a_2+\dots +a_{10}=62$, then $a_1$ is equal to :

If the sum of the first four terms of an A.P. is 6 and the sum of its first six terms is 4, then the sum of its first twelve terms is

Suppose $\mathrm{a},\mathrm{b},\mathrm{c}$ are in A.P. and ${\mathrm{a}}^2,2{\mathrm{b}}^2,{\mathrm{c}}^2$ are in G.P. If $\mathrm{a}<\mathrm{b}<\mathrm{c}$ and $\mathrm{a}+\mathrm{b}+\mathrm{c}=1$, then $9\left({\mathrm{a}}^2+{\mathrm{b}}^2+c^2\right)$ is equa …

Let ${\sum }_{\mathrm{k}=1}^{\mathrm{n}}{\mathrm{a}}_{\mathrm{k}}=\alpha {\mathrm{n}}^2+\beta \mathrm{n}$. If ${\mathrm{a}}_{10}=59$ and ${\mathrm{a}}_6=7{\mathrm{a}}_1$ then $\alpha +\beta$ is equal to

Let $729,81,9,1,\dots$ be a sequence and ${\mathrm{P}}_{\mathrm{n}}$ denote the product of the first $n$ terms of this sequence.

If $2{\sum }_{n=1}^{40}{\left(P_n\right)}^{\frac{1}{n}}=\frac{3^{\alpha }-1}{3^{\beta }}$ and $\gcd (\alpha ,\beta )=1$, then …

Consider an A.P.: ${\mathrm{a}}_1,{\mathrm{a}}_2,\dots .,{\mathrm{a}}_{\mathrm{n}};{\mathrm{a}}_1>0$. If ${\mathrm{a}}_2-{\mathrm{a}}_1=\frac{-3}{4},{\mathrm{a}}_{\mathrm{n}}=\frac{1}{4}{\mathrm{a}}_1$, and …

Let ${\alpha }_1,{\alpha }_2,{\alpha }_3,{\alpha }_4$ be an A.P. of four terms such that each term of the A.P. and its common difference $l$ are integers. If ${\alpha }_1+{\alpha }_2+{\alpha }_3+{\alpha }_4=48$ and ${\alpha }_1{\alpha }_2{\alpha }_3{\alpha }_4+l^4=361$ then t …

$\left(\frac{1}{3}+\frac{4}{7}\right)+\left(\frac{1}{3^2}+\frac{1}{3}\times \frac{4}{7}+\frac{4^2}{7^2}\right)+\left(\frac{1}{3^3}+\frac{1}{3^2}\times \frac{4}{7}+\frac{1}{3}\times \frac{4^2}{7^2}+\frac{4^3}{7^3}\right)+\dots$. upto infinite terms is equal …

The value of ${\sum }_{k=1}^{\infty }(-1{)}^{k+1}\left(\frac{k(k+1)}{k!}\right)$ is :

The common difference of the A.P.: ${\mathrm{a}}_1,{\mathrm{a}}_2,\dots .,{\mathrm{a}}_{\mathrm{m}}$ is 13 more than the common difference of the A.P.: ${\mathrm{b}}_1,{\mathrm{b}}_2,\dots ,{\mathrm{b}}_{\mathrm{n}}$. If ${\mathrm{b}}_{31}=-277,{\mathrm{b}}_{43}=-385$ an …

In a G.P., if the product of the first three terms is 27 and the set of all possible values for the sum of its first three terms is $\mathbb{R}-(\mathrm{a},\mathrm{b})$, then ${\mathrm{a}}^2+{\mathrm{b}}^2$ is equal to ______.

$\frac{6}{3^{26}}+\frac{10.1}{3^{25}}+\frac{10.2}{3^{24}}+\frac{10.2^2}{3^{23}}+\dots +\frac{10.2^{24}}{3}$ is equal to

If ${\sum }_{\mathrm{r}=1}^{25}\left(\frac{\mathrm{r}}{{\mathrm{r}}^4+{\mathrm{r}}^2+1}\right)=\frac{\mathrm{p}}{\mathrm{q}}$, where p and q are positive integers such that $\gcd (p,q)=1$, then $p+q$ is equal to ______。

Let A be the set of first 101 terms of an A.P., whose first term is 1 and the common difference is 5 and let B be the set of first 71 terms of an A.P., whose first term is 9 and the common difference is 7. Then the number of elements is $A\cap B$, which are divisible by 3, is :

If ${\sum }_{k=1}^n{a}_k=6n^3$, then ${\sum }_{k=1}^6{\left(\frac{a_{k+1}-a_k}{36}\right)}^2$ is equal to ______.

Let $\alpha =\frac{1}{4}+\frac{1}{8}+\frac{1}{16}+\dots \infty$ and
$\beta =\frac{1}{3}+\frac{1}{9}+\frac{1}{27}+\dots \infty$. Then the value of $(0.2{)}^{{\log }_{\sqrt{5}}(\alpha )}+(0.04{)}^{{\log }_5(\beta )}$ is equal to :

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 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 :

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

If the sum of the first 10 terms of the series $\frac{1}{1+1^4\times 4}+\frac{2}{1+2^4\times 4}+\frac{3}{1+3^4\times 4}+\frac{4}{1+4^4\times 4}+\dots$ is $\frac{\mathrm{m}}{\mathrm{n}},\gcd (\mathrm{m},\mathrm{n})=1$, then $\mathrm{m}+\mathrm{n}$ …

Let ${\mathrm{A}}_1,{\mathrm{A}}_2,{\mathrm{A}}_3,\dots \dots ..,{\mathrm{A}}_{39}$ be 39 arithmetic means between the numbers 59 and 159. Then the mean of ${\mathrm{A}}_{25},{\mathrm{A}}_{28},{\mathrm{A}}_{31}$ and ${\mathrm{A}}_{36}$ is equal to:

Let $\left(2^{1-\mathrm{a}}+2^{1+\mathrm{a}}\right),f(\mathrm{a}),\left(3^{\mathrm{a}}+3^{-\mathrm{a}}\right)$ be a A. P. and $\alpha$ be the minimum value of $f(\mathrm{a})$. Then the value of the integral …

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

The value of $1^3-2^3+3^3-\dots +15^3$ 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 …

For the functions $f(\theta )=\alpha {\tan }^2\theta +\beta {\cot }^2\theta$, and $\mathrm{g}(\theta )=\alpha {\sin }^2\theta +\beta {\cos }^2\theta ,\alpha >\beta >0$, let ${\min }_{0<\theta <\frac{\pi }{2}}f(\theta )={\max }_{0<\theta <\pi }\mathrm{g}(\theta )$. If the first term of a G. …

Let $\alpha =3+4+8+9+13+14+\dots$ 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

and $f(6)=37$. Then ${\sum }_{n=1}^{10}f(n)$ is equal to ____.

Let ${\mathrm{a}}_1,{\mathrm{a}}_2,{\mathrm{a}}_3,\dots \dots$ be an A.P. and ${\mathrm{g}}_1={\mathrm{a}}_1,{\mathrm{g}}_2,{\mathrm{g}}_3,\dots \dots$ be an increasing G.P. If ${\mathrm{a}}_1={\mathrm{a}}_2+{\mathrm{g}}_2=1$ and …

The sum $\frac{1^3}{1}+\frac{1^3+2^3}{1+3}+\frac{1^3+2^3+3^3}{1+3+5}+\dots$ up to 8 terms, is :

Let $a_1,a_2,a_3\dots$ 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

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 :

Suppose that the number of terms in an A.P. is 2 k , $\mathrm{k}\in \mathrm{N}$. 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 $3x^2-px+q=0$ are $10^{\text{th }}$ and $11^{\text{th }}$ terms of an arithmetic progression with common difference $\frac{3}{2}$. If the sum of the first 11 terms of this arithmetic progression is 88, then $q-2p$ is equal to ______.

Let $a_1,a_2,\dots ,a_{2024}$ be an Arithmetic Progression such that $a_1+(a_5+a_{10}+a_{15}+\dots +a_{2020})+a_{2024}=2233$. Then $a_1+a_2+a_3+\dots +a_{2024}$ is equal to

Consider an A.P. of positive integers, whose sum of the first three terms is 54 and the sum of the first twenty terms lies between 1600 and 1800. Then its $11^{\text{th}}$ term is :

The value of ${\lim }_{n\to \infty }\left({\sum }_{K=1}^n\frac{k^3+6k^2+11k+5}{(k+3)!}\right)$ is :

For positive integers $n$, if $4a_n=\left(n^2+5n+6\right)$ and ${\mathrm{S}}_{\mathrm{n}}={\sum }_{\mathrm{k}=1}^{\mathrm{n}}\left(\frac{1}{{\mathrm{a}}_{\mathrm{k}}}\right)$, then the value of $507{\mathrm{S}}_{2025}$ is :

If ${\sum }_{r=1}^{13}\left\{\frac{1}{\sin \left(\frac{\pi }{4}+(r-1)\frac{\pi }{6}\right)\sin \left(\frac{\pi }{4}+\frac{r\pi }{6}\right)}\right\}=a\sqrt{3}+b$, $\mathrm{a},\mathrm{b}\in \mathbf{Z}$, then ${\mathrm{a}}^2+{\mathrm{b}}^2$ is equal to :

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

If $f(x)=\frac{2^x}{2^x+\sqrt{2}},x\in R$, then ${\sum }_{k=1}^{81}f\left(\frac{k}{82}\right)$ is equal to :

Let $<{\mathrm{a}}_{\mathrm{n}}>$ be a sequence such that ${\mathrm{a}}_0=0,{\mathrm{a}}_1=\frac{1}{2}$ and $2{\mathrm{a}}_{\mathrm{n}+2}=5{\mathrm{a}}_{\mathrm{n}+1}-3{\mathrm{a}}_{\mathrm{n}},\mathrm{n}=0,1,2,3,\dots$. Then ${\sum }_{\mathrm{k}=1}^{100}{\mathrm{a}}_{\mathrm{k}}$ is …

Let ${\mathrm{T}}_{\mathrm{r}}$ be the ${\mathrm{r}}^{\text{th }}$ term of an A.P. If for some m , ${\mathrm{T}}_{\mathrm{m}}=\frac{1}{25},{\mathrm{T}}_{25}=\frac{1}{20}$ and $20{\sum }_{\mathrm{r}=1}^{25}{\mathrm{T}}_{\mathrm{r}}=13$, then …

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

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

Let ${\mathrm{S}}_{\mathrm{n}}=\frac{1}{2}+\frac{1}{6}+\frac{1}{12}+\frac{1}{20}+\dots$ upto n terms. If the sum of the first six terms of an A.P. with first term -p and common difference p is $\sqrt{2026{\mathrm{S}}_{2025}}$, then the absolute difference between …