Divisibility Induction

Question

Hey guys had a induction problem for hw and was having trouble with it. I tried mods and a/b where a and b are integers but cant seem to prove the last step.

Answer 1

Hint: $\;2^{n+1}+(-1)^n=2^{n+1}-(-1)^{n+1}\,$, then remember the identity $\,a^k-b^k=(a-b)(\cdots)\,$.

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[ EDIT ] For an alternative proof by induction, note that:

$$2^{n+1}+(-1)^n= 2 \cdot \left(2^n \color{blue}{+ (-1)^{n-1}}\right) \color{blue}{- 2 \cdot (-1)^{n-1}} + (-1)^n = 2 \cdot \left(2^n + (-1)^{n-1}\right) + 3 \cdot (-1)^n$$