Sigma n3

Q. The value of $\sum _{i=1}^n\sum _{j=1}^i\sum _{k=1}^{j}1=220,$ then the value of $n$ equals

1. $9$
2. $11$
3. $12$
4. $10$

Q. The difference between the sum of the first $k$ terms of the series $1^3+2^3+3^3+\dots \dots +n^3$ and the sum of the first $k$ terms of $1+2+3+\dots \dots n$ is $1980$. The value of $k$ is

Q.

The sum of the series $1^2$.2 + $2^2$.3 + $3^2$.4 + ........ to n terms is

1. $\frac{n^3{(n+1)}^3(2n+1)}{24}$

2. $\frac{n(n+1)(3n^2+7n+2)}{12}$

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

4. $\frac{n(n+1)}{12}$[6n(n+1) + 2(2n + 1)]

Q. If $3k,$ $k,$ $[k^2-34]$ are the first three terms of a geometric progression, where $k\in R^{+}$ and $[.]$ is the greatest integer function, then the value of $\sum _{r=1}^{10}(r{)}^{k/2}$ is

1. $55$
2. $385$
3. $3025$
4. $325$

Q. The sum of first 9 terms of the series.
$\frac{1^3}{1}+\frac{1^3+2^3}{1+3}+\frac{1^3+2^3+3^3}{1+3+5}+...$

1. 142
2. 192
3. 71
4. 96

Q.

If the sum of the $3^3+7^3+{11}^3+{15}^3+...\text{upto}20$ terms is $S_{20}$. Then the value of $S_{20}$ is___

Q. The sum of the series ${31}^3+{32}^3+\dots \dots +{50}^3$ is

1. $98092$
2. $121023$
3. $150569$
4. $1409400$

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. $27$
3. $54$
4. $108$

Q. If $f\left(x\right)=$$\{\begin{array}{c}|x|-3,x<1\\ |x-2|+a,x\ge 1\end{array},$ $g\left(x\right)=\{\begin{array}{c}2-|x|,x<2\\ \text{sgn}\left(x\right)-b,x\ge 2\end{array}$ and $h\left(x\right)=f\left(x\right)+g\left(x\right)$ is discontinuous at exactly one point, then which of the following values of $a$ and $b$ are possible

1. $a=-3,b=0$
2. $a=1,b=2$
3. $a=2,b=1$
4. $a=-3,b=2$

Q. The sum to $n$ terms of the series $(1\times 2\times 3)+(2\times 3\times 4)+(3\times 4\times 5)+\dots$ is

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