Question

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

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

Answer 1

The correct option is B

$i-1$

Explanation for the correct option:

Find the required value.

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

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

$$\begin{gathered} =(1+i)\sum _{n=1}^{13}{i}^n \\ =(1+i)(i+i^2+i^3+i^4+i^5+i^6+i^7+i^8+i^9+i^{10}+i^{11}+i^{12}+i^{13}) \\ =(1+i)(i-1-i+1+i-1-i+1+i-1-i+1+i) \\ =(1+i)i \\ =i+i^2 \\ =i-1\left[\because i^2=-1\right] \end{gathered}$$

Therefore, the value of the sum $\sum _{n=1}^{13}(i^n+i^{n+1})$ is $i-1$.

Hence, option (B) is the correct answer.