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

Answer 1

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.

Answer 2

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

Answer 3

Huh. Easier than I thought.