Sequences & Series · JEE Main previous year questions

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 :

1. A.$7360$
2. B.$7220$
3. C.$7260$
4. D.$7380$

The sum to 20 terms of the series $2\cdot 2^2-3^2+2\cdot 4^2-5^2+2\cdot 6^2-\dots \dots$ is equal to _________.

Let $x_1,x_2,\dots ,x_{100}$ be in an arithmetic progression, with $x_1=2$ and their mean equal to 200 . If $y_i=i(x_i-i),1\le i\le 100$, then the mean of $y_1,y_2,\dots ,y_{100}$ is :

1. A.10051.50
2. B.10049.50
3. C.10100
4. D.10101.50

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

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

1. A.241
2. B.231
3. C.220
4. D.210

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

Suppose $a_1,a_2,2,a_3,a_4$ 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 $\frac{49}{2}$, then $a_4$ is equal to _________.

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

1. A.$11280$
2. B.$11290$
3. C.$11310$
4. D.$11260$

Let $0<z<y<x$ be three real numbers such that $\frac{1}{x},\frac{1}{y},\frac{1}{z}$ 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 $3(x+y+z{)}^2$ is equal to ___________.

Let $a_1,a_2,a_3,\dots ,a_n$ be n positive consecutive terms of an arithmetic progression. If $\text{d}>0$ is its common difference, then $$\underset{n\to \infty }{\lim }\sqrt{\frac{d}{n}}\left(\frac{1}{\sqrt{a_1}+\sqrt{a_2}}+\frac{1}{\sqrt{a_2}+\sqrt{a_3}}+\dots +\frac{1}{\sqrt{a_{n-1}}+\sqrt{a_n}}\right)\text{ is}$$

1. A.$\frac{1}{\sqrt{d}}$
2. B.1
3. C.0
4. D.$\sqrt{d}$

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

1. A.3420
2. B.3450
3. C.3250
4. D.3520

The sum of 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

1. A.$\frac{58}{111}$
2. B.$\frac{56}{111}$
3. C.$\frac{55}{111}$
4. D.$\frac{59}{111}$

Let $a_1=8,a_2,a_3,\dots ,a_n$ 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 ________.

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

1. A.$7$
2. B.$14$
3. C.$3$
4. D.$\frac{9}{2}$

Let $a_1,a_2,\dots ,a_n$ be in A.P. If $a_5=2a_7$ and $a_{11}=18$, then $12\left(\frac{1}{\sqrt{a_{10}}+\sqrt{a_{11}}}+\frac{1}{\sqrt{a_{11}}+\sqrt{a_{12}}}+\dots +\frac{1}{\sqrt{a_{17}}+\sqrt{a_{18}}}\right)$ is equal to ___________.

Let $a_1,a_2,a_3,\dots$ be an A.P. If $a_7=3$, the product $a_1{a}_4$ is minimum and the sum of its first $n$ terms is zero, then $n!-4a_{n(n+2)}$ is equal to :

1. A.24
2. B.$\frac{381}{4}$
3. C.9
4. D.$\frac{33}{4}$

Let $a_1,a_2,a_3,\dots$ 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 ____.

Let $\{a_k\}$ and $\{b_k\},k\in N$, be two G.P.s with common ratios $r_1$ and $r_2$ respectively such that $a_1=b_1=4$ and $r_1<r_2$. Let $c_k=a_k+b_k,k\in N$. If $c_2=5$ and $c_3=\frac{13}{4}$ then $\sum _{k=1}^{\infty }{c}_k-(12a_6+8b_4)$ is equal to ____.

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 $7^{\mathrm{th}},9^{\mathrm{th}},17^{\mathrm{th}}$ terms of ${\mathrm{A}}_1,{\mathrm{A}}_2,{\mathrm{A}}_3$, respectively, such that $$\left|\begin{array}{ccc}a& 7& 1\\ 2b& 17& 1\\ c& 17& 1\end{array}\right|+70=0.$$ If $a=29$, then the sum of first 20 terms of an AP whose first term is $c-a-b$ and common difference is $\frac{d}{12}$, is equal to ____.

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

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

1. A.12
2. B.$-6$
3. C.6
4. D.2

The $4^{\text{th}}$ term of GP is $500$ and its common ratio is $\frac{1}{m},m\in \mathbb{N}$. Let ${\mathrm{S}}_n$ 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 of m 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 ____.

Let $\alpha ,\beta$ and $\gamma$ be three positive real numbers. Let $f(x)=\alpha x^5+\beta x^3+\gamma x,x\in \mathbf{R}$ and $g:\mathbf{R}\to \mathbf{R}$ be such that $g(f(x))=x$ for all $x\in \mathbf{R}$. If $a_1,a_2,a_3,\dots ,a_n$ be in arithmetic progression with mean zero, then the value of $f\left(g\left(\frac{1}{n}\sum _{i=1}^{n}f(a_i)\right)\right)$ is equal to :

1. A.0
2. B.3
3. C.9
4. D.27

For $\mathrm{p},\mathrm{q}\in \mathbf{R}$, consider the real valued function $f(x)=(x-\mathrm{p}{)}^2-\mathrm{q},x\in \mathbf{R}$ and $\mathrm{q}>0$. Let ${\mathrm{a}}_1,{\mathrm{a}}_2,{\mathrm{a}}_3$ and ${\mathrm{a}}_4$ be in an arithmetic progression with mean $\mathrm{p}$ and positive common difference. If $|f({\mathrm{a}}_i)|=500$ for all $i=1,2,3,4$, then the absolute difference between the roots of $f(x)=0$ is ____.