Sum of π∑cos(n)/5^n using only real numbers

[waealu]

Homework Statement

I am supposed to find the value of the infinite series:

$$\sum_{n=0}^{+\infty}\frac{\pi\cos(n)}{5^n}$$

Homework Equations

I asked this question before on this forum and micromass told me that I should use cos(n)=((e^i)^n+(e^(-i))^n)/2. That equation worked and I was able to find the correct answer. I used that equation to find a "a" and an "r" in order to find the solution using =a/(1-r).

But my instructor said he would like me to try and solve it by only using real numbers.

How would I start to solve that without the imaginary numbers?

Thanks

 

[lanedance]

just an idea, but how about considering angle sum formulas...
ie what is
cos(n)
cos(2n)
cos(3n)
in terms of only n?

 

[tiny-tim]

try multiplying by cos(1) …

S*cos(1) = ∑ π cos(n)cos(1)/5ⁿ = … ? ​