The value of ∑_n=1^13 (i^n+i^n+1 ), i=√(-1) is

Correct answer: (B) i-1. We have. ∑_n=1^13 (i^n+i^n+1 ) = ∑_n=1^13 i^n+∑_n=1^13 i^n+1 = i (frac1-i^131-i )+i^2 (frac1-i^131-i ) = i ((1-i)/(1-i) )- ((1-i)/(1-i) ) = i-1

Q51 / 155 · WBJEE 2016 · Mathematics

The value of $\sum_{n=1}^{13}\left(i^{n}+i^{n+1}\right), i=\sqrt{-1}$ is

A
$i$
Correct answer
B
$i-1$
C
1
D
0

Solution

We have. $\begin{aligned} & \sum_{n=1}^{13}\left(i^{n}+i^{n+1}\right) \\=& \sum_{n=1}^{13} i^{n}+\sum_{n=1}^{13} i^{n+1} \\=& i \left(\frac{1-i^{13}}{1-i}\right)+i^{2}\left(\frac{1-i^{13}}{1-i}\right) \\=& i \left(\frac{1-i}{1-i}\right)-\left(\frac{1-i}{1-i}\right) \\=& i-1 \end{aligned}$

Hence, the correct answer is option: 'B'.

The value of $\sum_{n=1}^{13}\left(i^{n}+i^{n+1}\right), i=\sqrt{-1}$ is

Solution

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