How do I prove for any natural number (n), this expression is divisible by 10?

(3^(n+2)-2^(n+2)+3^n-2^n)/10

Question

How do I prove for any natural number (n), this expression is divisible by 10?

(3^(n+2)-2^(n+2)+3^n-2^n)/10

anonymous · Answer

If you're proving for any natural number, then you're probably going to end up using a proof by induction. The base case is trivial, I haven't attempted the induction step yet.

zarkon · Answer

$$3^{n+2}-2^{n+2}+3^n-2^n$$
$$=3^{n}3^2-2^{n}2^2+3^n-2^n$$
$$=3^{n}9-2^{n}4+3^n-2^n$$
$$=10\cdot3^{n}-5\cdot 2^{n}$$
$$=10\cdot3^{n}-10\cdot 2^{n-1}$$
$$=10(3^{n}- 2^{n-1})$$

anonymous · Answer

Huh. Easier than I thought.