A divisibility problem with mathematical induction

Question

Prove by mathematical induction that if $k$ is odd and $n$ is a natural number, then $2^{n + 2}$ divides $k^{2n} - 1$.

I'm stuck while assuming $n = q$ is true as hypothesis, as I can't prove for $q + 1$.

Answer 1

I think the correct problem is if $k^{2^q}=1+2^qr$

$k^{2^{q+1}}=(k^{2^q})^2=(1+2^qr)^2\equiv1\pmod{2^{q+1}}$