Everything Can Be Cancelled Off?

Algebra Level 3

$\large i + i^2 + i^3 + i^4 + \cdots + i^n = - 1$

The equation above holds true for some positive integer $n$ . Which of the following is a possible value of $n$ ?

Clarification : $i=\sqrt{-1}$ .

15 17 16 18

1 solution

Siddharth Singh
Mar 30, 2016

We first see first 4 terms,

$i+i^{2}+i^{3}+i^{4}=i+(-1)+(-i)+1=0$ (if $n=4k$ the terms cancel each other,which gives 0)

To get $-1$ we need to have $n=4k-1$

Since, $15=4k-1$ where $k=4$ ,hence $15$ is the answer.