Question

Write the sum to n terms of a series whose $r^{th}$ term is :r + $2^r$.

Answer 1

Let $T_n$ be the given series and $S_n$ be the sum to n terms of the given series. $\therefore T_n=r+2^r\therefore S_n={\sum }_{r-1}^n(r+2^r)={\sum }_{r-1}^{n}r+{\sum }_{r-1}^n{2}^r=\frac{n(n+1)}{2}+[2^1+2^2..{.2}^n]=\frac{n(n+1)}{2}+\frac{2(2^n-1)}{(2-1)}=\frac{n(n+1)}{2}+\frac{2\times 2^n-2}{1}=\frac{n(n+1)}{2}+2^{n+1}-2\therefore S_n=\frac{n(n+1)}{2}+2^{n+1}-2$