<s>[INST] Proving that $2^{2\cdot 3^{n-1}}\equiv 1+3^n\pmod{3^{n+1}}$ for every natural $n$ 
Prove that $2^{2\cdot 3^{n-1}}\equiv 1+3^n\pmod{3^{n+1}}$ for every natural $n$

Of course, should be done by induction. Base case ($n=1$) is easy.
I got stuck with the step:
Let's assume that it's right for $n$. Then,
$$2^{2\cdot 3^{n-1}} = 1 + 3^n \pmod{3^{n+1}}$$
We want to evaluate $$2^{2 \cdot 3^n} \pmod {3^{n+2}}$$
I think we can somehow utilize the fact that $2,3$ are coprime and reduce $3^{n+2}$ to $3^{n+1}$. 
I'd be glad for help on that.
Thanks!
 [/INST] $(1+3^nx)^3 = 1+3^{n+1}x+3^{2n+1}x^2+3^{3n}x^3\equiv 1+3^{n+1}x \pmod{3^{n+2}}$ if $n\ge 1$.
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