Sequences & Series · JEE Main previous year questions

If the sum of the series $1^2+2\cdot 2^2+3^2+2\cdot 4^2+5^2+2\cdot 6^2+\dots$ upto $n$ terms, when $n$ is even, is $\frac{n(n+1{)}^2}{2}$, then the sum of the series, when $n$ is odd, is

1. A.$n^2(n+1)$
2. B.$\frac{n^2(n-1)}{2}$
3. C.$\frac{n^2(n+1)}{2}$
4. D.$n^2(n-1)$

The sum of the series $\frac{1}{1+\sqrt{2}}+\frac{1}{\sqrt{2}+\sqrt{3}}+\frac{1}{\sqrt{3}+\sqrt{4}}+\dots$ upto 15 terms is

1. A.1
2. B.2
3. C.3
4. D.4

The difference between the fourth term and the first term of a Geometrical Progression is 52. If the sum of its first three terms is 26 , then the sum of the first six terms of the progression is

1. A.63
2. B.189
3. C.728
4. D.364

The sum of the series $1^2+2\cdot 2^2+3^2+2\cdot 4^2+5^2+2\cdot 6^2+\cdots +2(2m{)}^2$ is

1. A.$m(2m+1{)}^2$
2. B.$m^2(m+2)$
3. C.$m^2(2m+1)$
4. D.$m(m+2{)}^2$

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

1. A.$34\mathrm{minutes}$
2. B.$125\mathrm{minutes}$
3. C.$135\mathrm{minutes}$
4. D.$24\mathrm{minutes}$

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

1. A.3
2. B.4
3. C.6
4. D.2

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

1. A.$\frac{41}{11}$
2. B.$\frac{7}{2}$
3. C.$\frac{2}{7}$
4. D.$\frac{11}{41}$

If $a_1,a_2,\dots a_n$ are in H.P., then the expression $a_1{a}_2+a_2{a}_3+\cdots +a_{n-1}{a}_n$ is equal to

1. A.$n(a_1-a_n)$
2. B.$(n-1)(a_1-a_n)$
3. C.$n{a}_1{a}_n$
4. D.$(n-1)a_1{a}_n$

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

1. A.G.P.
2. B.A.P.
3. C.Arithmetic-Geometric Progression
4. D.H.P.

The sum of the series $1+\frac{1}{4.2!}+\frac{1}{16.4!}+\frac{1}{64.6!}+\dots \dots$ ad inf. is

1. A.$\frac{e-1}{\sqrt{e}}$
2. B.$\frac{e+1}{\sqrt{e}}$
3. C.$\frac{e-1}{2\sqrt{e}}$
4. D.$\frac{e+1}{2\sqrt{e}}$

Let $T_r$ be the rth term of an A.P. whose first term is a and common difference is d. If for some positive integers m, n, $m\ne n$, $T_m=\frac{1}{n}$ and $T_n=\frac{1}{m}$, then a - d equals

1. A.$\frac{1}{m}+\frac{1}{n}$
2. B.1
3. C.$\frac{1}{mn}$
4. D.0

The sum of the first n terms of the series $1^2+2.2^2+3^2+2.4^2+5^2+2.6^2+\dots$ is $\frac{n(n+1{)}^2}{2}$ when n is even. When n is odd the sum is

1. A.${\left[\frac{n(n+1)}{2}\right]}^2$
2. B.$\frac{n^2(n+1)}{2}$
3. C.$\frac{n(n+1{)}^2}{4}$
4. D.$\frac{3n(n+1)}{2}$

The sum of series $\frac{1}{2!}+\frac{1}{4!}+\frac{1}{6!}+\dots \dots$ is

1. A.$\frac{(e^2-2)}{e}$
2. B.$\frac{(e-1{)}^2}{2e}$
3. C.$\frac{(e^2-1)}{2e}$
4. D.$\frac{(e^2-1)}{2}$

If 1, ${\log }_9(3^{1-x}+2)$, ${\log }_3(4\cdot 3^x-1)$ are in A.P. then x equals

1. A.${\log }_{3}4$
2. B.$1-{\log }_{3}4$
3. C.$1-{\log }_{4}3$
4. D.${\log }_{4}3$

l, m, n are the $p^{th}$, $q^{th}$ and $r^{th}$ term of a G.P all positive, then $\left|\begin{array}{ccc}\log l& p& 1\\ \log m& q& 1\\ \log n& r& 1\end{array}\right|$ equals

1. A.$-1$
2. B.$2$
3. C.$1$
4. D.$0$

Fifth term of a GP is 2, then the product of its first 9 terms is

1. A.256
2. B.512
3. C.1024
4. D.none of these

Sum of infinite number of terms of GP is 20 and sum of their square is 100. The common ratio of GP is

1. A.5
2. B.3/5
3. C.8/5
4. D.1/5