$\displaystyle 5^{2n}-3^{n}$ is a multiple of 11 for all integers $\displaystyle n\geq 1$

My attempt at a solution so far (I haven't included shown the case for P(1) etc.

$\displaystyle P(k):5^{2k}-3^{k}=11p \\ P(k+1):5^{2k+2}-3^{k+1} \\ 5^{2}\cdot 5^{2k}-3\cdot 3^{k} \\5^{2}( 3^{k}+11p)-3\cdot3^{k}$

I am not sure if it's correct up to here, and what the next step is.

I have done the steps for P(1) to verify when n=1 (just not shown).

Thank you for any help.

P.S, I'm not used to using LaTeX