# Difference between revisions of "1981 AHSME Problems/Problem 24"

Alexwin0806 (talk | contribs) (→Solution) |
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<cmath>x=\cos(\theta) + \sqrt{\cos^2(\theta)-1}</cmath> | <cmath>x=\cos(\theta) + \sqrt{\cos^2(\theta)-1}</cmath> | ||

− | <cmath>x=\cos(\theta) + \sqrt{(-1)(\sin^2(\theta | + | <cmath>x=\cos(\theta) + \sqrt{(-1)(\sin^2(\theta)}</cmath> |

<cmath>x=\cos(\theta) + i\sin(\theta)</cmath> | <cmath>x=\cos(\theta) + i\sin(\theta)</cmath> | ||

## Revision as of 21:19, 1 May 2020

## Problem

If is a constant such that and , then for each positive integer , equals

## Solution

Multiply both sides by and rearrange to . Using the quadratic equation, we can solve for . After some simplifying:

Substituting this expression in to the desired gives:

Using DeMoivre's Theorem:

Because is even and is odd:

Which gives the answer