Question

The value of the sum $\sum _{n=1}^{13}\left(i^n+i^{n+1}\right),$ where $i$ is the complex number $\sqrt{-1,}$ is -.

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

Answer 1

The correct option is D $i-1$

$i^n=\{\begin{array}{c}1,n=4k\\ i,n=4k+1\\ -1,n=4k+2\\ -i,n=4k+3\end{array}$

& $i^{4k}+i^{4k+1}+i^{4k+2}+i^{4k+3}=0$

$i^n+i^{n+1}=i^n(i+1)$

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

$=(i+1)(i+i^2+i^3+....+i^{13})$

$=(i+1)(0+0+0+i^{13})=(i+1)\left(i\right)=i-1$