Sigma n2

Q.

Complete the series $4,5,12,39,160,.....$

Q. Let $S_n=1\cdot (n-1)+2\cdot (n-2)+3\cdot (n-3)+\cdots +(n-1)\cdot 1,n\ge 4.$ The sum $\sum _{n=4}^{\infty }(\frac{2S_n}{n!}-\frac{1}{(n-2)!})$ is equal to

1. $\frac{e-1}{3}$
2. $\frac{e}{3}$
3. $\frac{e}{6}$
4. $\frac{e-2}{6}$

Q. For the series, $S=1+\frac{1}{(1+3)}(1+2{)}^2+\frac{1}{(1+3+5)}(1+2+3{)}^2+\frac{1}{(1+3+5+7)}(1+2+3+4{)}^2+\dots \dots$

1. $7$th term is $16$
2. $7$th term is $18$
3. Sum of first $10$ terms is $\frac{505}{4}$
4. Sum of first $10$ terms is $\frac{405}{4}$

Q.

${lim}_{n\to \infty }\frac{1^2+2^2+3^2\cdots n^2}{n^3}$ is equal to

1. $\frac{1}{2}$

2. $\frac{1}{3}$

3. 1

4. 0

Q. The sum of $(11{)}^2+(12{)}^2+(13{)}^2+\dots +(20{)}^2$ is

1. $3485$
2. $2555$
3. $2885$
4. $2485$

Q. The sum of the series $1\times n+2(n-1)+3(n-2)+\dots \dots +(n-1)\times 2+n\times 1$ is

1. $\frac{n(n+1)(n+2)}{6}$
2. $\frac{n(2n+1)(n+1)}{4}$
3. $\frac{n(n+1)(n+2)(n+3)}{6}$
4. $\frac{n(2n+1)(n+1)(n+2}{4}$

Q.

4 + 6 + 9 + 13 + 18 +....

Q. Let $\alpha =\frac{-1+i\sqrt{3}}{2}$.
If $a=(1+\alpha )\sum _{k=0}^{100}{\alpha }^{2k}$ and $b=\sum _{k=0}^{100}{\alpha }^{3k}$,
then $a$ and $b$ are the roots of the quadratic equation:

1. $x^2+101x+100=0$
2. $x^2+102x+101=0$
3. $x^2-102x+101=0$
4. $x^2-101x+100=0$

Q. The sum to $n$ terms of the series $\frac{1}{1+1^2+1^4}+\frac{2}{1+2^2+2^4}+\frac{3}{1+3^2+3^4}+\dots \dots$ is $\frac{1}{2}-\frac{1}{a(n^b+n^c+d{n}^4+1)}$, then $a+b+c+d=$

1. $5$
2. $8$
3. $4$
4. $6$

Q. Sum up to $16$ terms of the series $\frac{1^3}{1}+\frac{1^3+2^3}{1+2}+\frac{1^3+2^3+3^3}{1+2+3}+\dots \dots$ is

1. $850$
2. $856$
3. $816$
4. $842$

Q. The value of $\underset{-1/\sqrt{2}}{\overset{1/\sqrt{2}}{\int }}{({\left(\frac{x+1}{x-1}\right)}^2+{\left(\frac{x-1}{x+1}\right)}^2-2)}^{1/2}dx$ is

1. $2{\log }_{e}16$
2. ${\log }_{e}16$
3. $4{\log }_e(3+2\sqrt{2})$
4. ${\log }_{e}4$

Q. Let $S_k=\frac{1+2+3+\cdots +k}{k}.$ If
$S_1^2+S_2^2+\cdots +S_{19}^2=\frac{A}{4},$ then the value of $A$ is

Q. Let $f\left(n\right)$ denote the $n^{th}$ term of the sequence $3,6,11,18,27,...$ and $g\left(n\right)$ denote the $n^{th}$ term of the sequence $3,7,13,21,...$ . Let $F\left(n\right)$ and $G\left(n\right)$ denote the sum of $n$ terms of the above sequences, respectiveley. $\underset{n\to \infty }{lim}\frac{F\left(n\right)}{G\left(n\right)}=$

1. $2$
2. $1$
3. $0$
4. $\infty$

Q. If $|x|<1,|y|<1$ and $x\ne y,$ then the sum to infinity of the following series $(x+y)+(x^2+xy+y^2)+(x^3+x^{2}y+x{y}^2+y^3)+......\infty$

1. $\frac{x+y+xy}{(1-x)(1-y)}$
2. $\frac{x+y-xy}{(1-x)(1-y)}$
3. $\frac{x+y+xy}{(1+x)(1+y)}$
4. $\frac{x+y-xy}{(1+x)(1+y)}$

Q. Find the sum to $n$ terms of the series: $5+11+19+29+41\dots$

Q.

Find a formula for the general term of the sequence, assuming that the pattern of the first few terms continues. $($Assume that $n$ begins with $1)$.

$1,\frac{1}{3},\frac{1}{5},\frac{1}{7},\frac{1}{9},...$ then $a_n=\_\_\_\_\_\_$

Q. For the series, $S=1+\frac{1}{(1+3)}(1+2{)}^2+\frac{1}{(1+3+5)}(1+2+3{)}^2+\frac{1}{(1+3+5+7)}(1+2+3+4{)}^2+....$

1. $7\text{th}$ term is $16$
2. $11\text{th}$ term is $32$
3. sum of first $7$ terms is $\frac{203}{4}$
4. sum of first $10$ terms is $\frac{505}{4}$

Q. If $P\left(n\right):{}^{\prime \prime }{2}^{2n}-1$ is divisible by $k$ for all $n\in N^{\prime \prime }$ is true, then the value of ${}^{\prime }{k}^{\prime }$ is

1. 6
2. 3
3. 7
4. 2

Q. Find the sum of the following series up to $n$ terms:
$5+55+555+...$
$.6+.66+.666+...$

Q. Describe the following set in Roster form:

(iv) $\{x\in N:x=2n,n\in N\}$

Q. If $p$ denoted the fractional part of the number $p$, then $\left\{\frac{3^{200}}{8}\right\},$ is equal to:

1. $\frac{5}{8}$
2. $\frac{1}{8}$
3. $\frac{7}{8}$
4. $\frac{3}{8}$

Q. Let $P\left(x\right)$ be a four-degree polynomial with leading coefficient equal to $1.$ If $P\left(1\right)=-1,$ $P\left(2\right)=1,$ $P\left(3\right)=3,$ $P\left(4\right)=5,$ then $\underset{x\to 1}{lim}\frac{P\left(x\right)+3-2x}{(x-1)}$ is equal to

1. $6$
2. $5$
3. $2$
4. $-6$

Q. Find the sum of the following series to n terms : (1-7) $1^3+3^3+5^3+7^3+....$

Q. Sum to infinity of the series $1+\frac{4}{5}+\frac{7}{5^2}+\frac{10}{5^3}+...=$

1. $\frac{16}{35}$
2. $\frac{35}{16}$
3. $\frac{1}{8}$
4. $\frac{35}{8}$

Q.

Sum of first n terms in the following series

${cot}^{-1}$3 + ${cot}^{-1}$7 + ${cot}^{-1}$13 + ${cot}^{-1}$21 + .........n terms.is given by

1. ${tan}^{-1}$( $\frac{n}{n+2}$)

2. ${cot}^{-1}$( $\frac{n+2}{n}$)

3. ${tan}^{-1}$(n+1) - ${tan}^{-1}$ 1

4. All of these

Q.

If the sum of n terms of an A.P. is cn(n-1), where c$\ne$0, then sum of the squares of these terms is

1. none of these

2. 

3. 

4. 

Q. Sum to n terms the series $1^2-2^2+3^2-4^2+5^2-6^2+.........$

Q. The value of $\underset{2}{\overset{8}{\int }}|x-5|dx$ is

Q. If $n$ is odd, then the sum of $n$ terms of the series $1^2-2^2+3^2-4^2+5^2-6^2+...$ is

1. $\frac{n(n-1)}{2}$
2. $\frac{n(n+1)}{2}$
3. $\frac{-n(n-1)}{2}$
4. $\frac{-n(n+1)}{2}$

Q.

If $\sum _{i=1}^{\pi }i$ = $\frac{i(i+1)}{2}$, then $\sum _{i=1}^n(3i-2)$ =