[Math] Proof by induction: Show that $9^n-2^n$ for any $n$ natural number is divisible by $7$.

Question

Can someone please solve following problem.

Show that $9^n-2^n$ for any $n$ natural number is divisible by $7$. ($9 ^ n$ = $9$ to the power of $n$).

I know the principle of induction but am stuck with setting up a formula for this.

Answer 1

Hint:

Assume that $7\mid (9^n-2^n)$,

For $n+1:$

$$9^{n+1}-2^{n+1}=9^n9-2^n2=9^n(7+2)-2^n2=9^n7+9^n2-2^n2=9^n7+2(9^n-2^n)$$