How do I prove that powers of 5 are the sum of two squares using mathematical induction

Continuing with your induction, assuming that $5^n=a_n^2+b_n^2$ then multiplying both sides by $5^2$ gives $5^{n+2}=(5a_n)^2+(5b_n)^2$. So $5^{n+2}$ is a sum of two squares if $5^{n}$ is. We know it is for $n=2$, so it must be for $n=2+2=4$, etc. Hence by induction, for all even $n \geq 2$ we have that $5^n$ is a sum of two squares. We also know $5^n$ is expressible as the sum of two squares for $n=1$, so it must be for $n=1+2=3$ and by induction for all odd $n \geq 1$. Because a positive integer is either even or odd, we have accounted for all positive integers.

Hint: $$(x^2+y^2)(s^2+t^2) = (xs-yt)^2 + (xt+ys)^2 $$