Question

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

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

Answer 1

The correct option is D $i-1$

$\sum _{n=1}^{13}(i^n+i^{n+1})$

$=(i+1)\sum _{n=1}^{13}{i}^n$

$=(i+1)\frac{\left(i\right(i^{13}-1\left)\right)}{i-1}$

$=(i+1)\frac{\left(i\right(i-1\left)\right)}{i-1}$

$=(i+1)\left(i\right)$

$=i-1$
Hence, the correct answer is D.