Question

The value of the sum $\sum _{n=1}^{10}(i^n+i^{n+1})$, where $i=\sqrt{-1}$, equals

• A. $i$
• B. $-2$
• C. $-i$
• D. $0$

Answer 1

Answer: (B)

$-2$

Expanding the series we get
$i+i^2+i^2+i^3+i^3+i^4...+i^{10}+i^{10}+i^{11}$
$=(i+i^2+i^3...i^{10})+(i^2+i^3...i^{10}+i^{11})$
$=\frac{i(i^{10}-1)}{i-1}+\frac{-1(i^{10}-1)}{i-1}$
$=\frac{i^{10}-1}{i-1}[i-1]$
$=i^{10}-1$
$=(i^2{)}^5-1$
$=-1-1$
$=-2$
Alternatively
$i^n+i^{n+1}$
$=i^n(1+i)$
Summation implies
$(1+i)[i+i^2+i^3+...i^{10}]$
$=(1+i)\left[\frac{i(1-i^{10})}{1-i}\right]$
$=(1+i)\left[\frac{i(1-(-1\left)\right)}{1-i}\right]$
$=(1+i)\left[\frac{2i}{1-i}\right]$
$=\frac{2(i-1)}{1-i}$
$=-2$