Proof by induction: Show that $9^n-2^n$ for any $n$ natural number is divisible by $7$. 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. 
 A: Let $f(k)=9^k-2^k$ where $k \in \mathbb{N}$.
Base Case: $f(1)=9-2=7=7(1)$.
Inductive Hypothesis: Assume $f(k)$ is divisible by $7$ for some natural number $k$.
$f(k+1)= 9^{k+1}-2^{k+1} = 9\cdot9^k-2\cdot2^k$.
Inductive step: $f(k+1)-2f(k)=7\cdot9^k=7\cdot9^k$. So $f(k+1) =7\cdot 9^k+2f(k)$.
Recall: $f(k)$ was assumed to be divisible by $7$ for the inductive hypothesis. So we are done since $f(k+1)$ is the sum of two things divisible by $7$.
A: Hint:
Assume that $7\mid (9^n-2^n)$, 
For $n+1:$
$$9^{n+1}-2^{n+1}=9^n9-2^n2=9^n(7+2)-2^n2=9^n7+9^n2-2^n2=9^n7+2(9^n-2^n)$$
A: Hint: $x^n-y^n=(x-y)(x^{n-1}+yx^{n-2}+\dotsb+y^{n-2}x+y^{n-1})$ for $n\geq1$.
A: Assume $7\mid(9^n-2^n)$; then you can write $9^n-2^n=7m$, for some integer $m$, or $9^n=7m+2^n$ as well. Then
$$
9^{n+1}-2^{n+1}=
9\cdot 9^n-2^{n+1}=
9(7m+2^n)-2^{n+1}=
63m+(9-2)2^n=
7(9m+2^n)
$$
A: There shouldn't really be a formula for this problem, just setting up the induction step which usually involves plugging in $n = k+1$ into your expression. First show $7\mid 9^n-2^n$ for $n=1$. Then suppose the result holds for some $k \geq 1$ (i.e. you are allowed to assume $9^k-2^k$ is divisible by $7$.) Consider then the quantity $9^{k+1}-2^{k+1}$. Proving that this quantity must also be divisible by $7$ is all that remains. Doing a little bit of algebra manipulation will get the result. For example notice that  $$9^{k+1}-2^{k+1} = 9^{k}(7+2)-2^{k+1}$$ Can you proceed from here?
