R D Sharma Solutions for Chapter: Some Special Series, Exercise 3: VERY SHORT ANSWER QUESTIONS (VSAQs)

If the sum of the series: $2+4+6+8+\dots +2n$ is of the form $a{n}^x+b{n}^y$, then find the value of $a+x+b+y$.

If $1^2-2^2+3^2-4^2+5^2-6^2+\dots +(2n-1{)}^2-(2n{)}^2=k(2n+1)$, then what is the value of $k$.

If the sum to $n$ terms of a series whose $r^{\mathrm{th}}$ term is $r+2^r$ is of the form $\left(\frac{n^2+n}{k}+k^{n+1}-k\right)$, then write the value of $k$.

If$\sum _{r=1}^{n}r=55,$ find $\sum _{r=1}^n{r}^3$.

If the sum of first $n$ even natural numbers is equal to $k$ times the sum of first $n$ odd natural numbers, then write the value of $k$

Write the sum of $20$ terms of the series: $1+\frac{1}{2}(1+2)+\frac{1}{3}(1+2+3)+\dots$

Write the ${50}^{th}$ term of the series $2+3+6+11+18+\dots$

Let $S_n$ denote the sum of the cubes of first $n$ natural numbers and $s_n$ denote the sum of first $n$ natural numbers. Then, write the value of$\sum _{r=1}^n\frac{S_r}{s_r}$