Question

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

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

Answer 1

Answer: (B)

$i-1$

The value of $\sum _{n=1}^{13}(i^n+i^{n+1})$ is

$=(i+i^2+i^3+.....+i^{13})+(i^2+i^3+.....+i^{14})$

$=\frac{i(1-i^{13})}{1-i}+\frac{i^2(1-i^{13})}{1-i}$

$=\frac{i(1-i)}{1-i}+\frac{i^2(1-i)}{1-i}$

$=i+i^2=i-1$