Question

The value of the sum $\sum _{n=1}^{13}(i^n+i^{n+1})$, where $i=\sqrt{-1}$, equals

• A. $i$
• B. $i-1$
• C. $-i$
• D. $0$

Answer 1

The correct option is B $i-1$

Note $i^k+i^{k+1}+i^{k+2}+i^{k+3}=0$ .

Hence ${\sum }_{n=1}^{13}{i}^n=i^{13}=i$ and ${\sum }_{n=1}^{13}{i}^{n+1}=i^{14}=-1$.

Thus the given sum is $i-1$.