Question

What is the value of

begin mathsize 22px style i squared plus i to the power of 4 plus i to the power of 6 plus.... plus i to the power of 2 n end exponent space comma end style



where n is even number.

Answer 1

0

Explanation:

There is a pattern when raising i to positive integers: i² = -1, i³=-i and i^4 = 1. This repeats every 4 terms eg i^5 = i, i^6 = -1 and so on...

So if we look at i² + i^4 + i^6 + ... + i^(2n)

If n is even (a multiple of 2) then 2n must be a multiple of 4. So we know that the last term in the sequence will be 1 as i^4 = 1

So the sum is (-1) + 1 + (-1) + ... + 1

It ends on a multiple of 4 so there are an even number of terms in the sequence, hence -1 and 1 will cancel with each to get 0 as the answer.