Question

What is the value of the sum
$\sum _{n=2}^{11}(i^n+i^{n+1})$ where $i=\sqrt{-1}$?

• A. $i$
• B. $2i$
• C. $-2i$
• D. $1+i$

Answer 1

The correct option is C $-2i$

$\sum _{n=1}^{11}(i^n+i^{n+1})i=\sqrt{-1}$

Now $i^2=-1,i^3=-i,i^4=1,i^5=i$ after this values repeats

$i^2+i^3+i^4+i^5=0$

$\therefore \sum _{n=2}^{11}{i}^n=i^{11}+i^{12}=-i+1$

$\therefore \sum _{n=2}^{11}{i}^n+i^{n+1}=-2i$.