Question

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

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

Answer 1

Answer: (B)

$i-1$

$\sum _{n=1}^{13}(i^n+i^{n+1})=\sum _{n=1}^{13}{i}^n+\sum _{n=1}^{13}{i}^{n+1}$
$=i\left(\frac{1-i^{13}}{1-i}\right)+i^2\left(\frac{1-i^{13}}{1-i}\right)$
$=i\frac{(1-i)}{(1-i)}-\frac{(1-i)}{(1-i)}$
$=i-1$