Divisibility from index laws

In this question \(n\) is a positive integer.

1. Show that \(2^{n+2}-2^n\) is divisible by \(3\) for every \(n\).[2]
2. Show that \(5^{n+1}+5^n\) is divisible by \(6\) for every \(n\).[2]
3. Find the value of \(n\) for which \(3^{n+1}-3^{n-1}=216\).[3]
4. Explain why \(2^n+2^{n+1}\) is never a multiple of \(5\), however large \(n\) is.[2]