Sigma n2

Q. The sum $1+\frac{1^3+2^3}{1+2}+\frac{1^3+2^3+3^3}{1+2+3}+\cdot \cdot \cdot +\frac{1^3+2^3+3^3+\cdot \cdot \cdot +{15}^3}{1+2+3+\cdot \cdot \cdot +15}-\frac{1}{2}(1+2+3+\cdot \cdot \cdot +15)$ is equal to :

1. $620$
2. $660$
3. $1240$
4. $1860$

Q. If the sum of the first $15$ terms of the series
${\left(\frac{3}{4}\right)}^3+{\left(1\frac{1}{2}\right)}^3+{\left(2\frac{1}{4}\right)}^3+3^3+{\left(3\frac{3}{4}\right)}^3+....$

is equal to $225k$, then $k$ is equal to :

1. $9$
2. $54$
3. $27$
4. $108$

Q. The value of positive integer $n$ for which the quadratic equation, $\sum _{k=1}^n(x+k-1)(x+k)=10n$ has solutions $\alpha$ and $\alpha +1$ for some $\alpha$, is

Q. The sum of series $4-9x+16x^2-25x^3+36x^4-49x^5+\dots +\infty$ is

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

Q. Determine the second smallest prime factor of $\frac{1^3+1}{1+1}+\frac{2^3+1}{2+1}+\frac{3^3+1}{3+1}+\cdots +\frac{{2005}^3+1}{2005+1}.$
(correct answer + 3, wrong answer 0)

Q. If $S_n=\frac{1}{1\times 3}+\frac{1}{3\times 5}+\frac{1}{5\times 7}+\cdots n\text{terms}$, then $S_{\infty }$ is

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

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.

The value of the series $\left(\frac{1^1}{2}\right)+\left(\frac{1^2+2^2}{3}\right)+\left(\frac{1^2+2^2+3^2}{4}\right)+.....$ is

1. $\frac{n(n+1)(4n+5)}{36}$
2. $\frac{n(n-1)(4n+5)}{6}$
3. $\frac{n(n+1)(4n+2)}{3}$
4. None of these

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

1. 100
2. 99
3. 102
4. 101

Q. In the sum of first n terms of an A.P. is $c{n}^2$. then the sum of squares of these n terms is

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

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.

The sum of n terms of the series whose $n^{th}$ term is n(n+1) is equal to

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

2. $\frac{(n+1)(n+2)(n+3)}{12}$

3. $n^2$(n+2)

4. n(n+1)(n+2)

Q. Let $a,b,c\in R$. If $f\left(x\right)=a{x}^2+bx+c$ is such that $a+b+c=3$ and $f(x+y)=f\left(x\right)+f\left(y\right)+xy,\forall x,y\in R,$ then $\sum _{n=1}^{10}f\left(n\right)$ is equal to:

1. $330$
2. $165$
3. $190$
4. $255$

Q.

Find the sum of the series $2^2$ + $4^2$ + $6^2$ + $8^2$ +................ 20 terms ___