How to create an infinite Table of numbers

I wish to create to generate the set $ \{\pm \, 5^n : n\in \mathbb{N}\cup \{0\} \}$ and solve the equation $ a+b =c+d$ for all elements quadruplets $ a,b,c,d$ in the set which satisfy it.

How would I go about this problem. I know how to use the Table and Solve commands in general but not in this particular case