1. ## Proof

Prove by induction that

$2( 4^{2n+1}) + 3^{3n+1}
$

is divisible by 11 $\forall n \! \in \! \mathbb{N}$
..

2. are you sure?
for n = 1, $4^{2(1)+1} + 3^{3(1)+1} = 4^3 + 3^4 = 145$

3. Sorry the latex was a bit confusing, I cleaned it up.

4. Ok, let's assume it holds for n, we'll now show it does so for n+1

$a_{n}=2\cdot{4^{2n+1}}+3^{3n+1}$ is divisible by 11

$a_{n+1}=2\cdot{4^{2n+3}}+3^{3n+4}$

$a_{n}-2\cdot{4^{2n+1}}=3^{3n+1}$ then $3^{3}\cdot(a_{n}-2\cdot{4^{2n+1}})=3^{3n+4}$

So: $a_{n+1}=2\cdot{4^{2n+3}}+27\cdot{a_{n}}-54\cdot{4^{2n+1}}$

Since $2\cdot{4^{2n+3}}-54\cdot{4^{2n+1}}=2\cdot{4^{2n+1}}\cdot{(16-27)}=-2\cdot{4^{2n+1}}\cdot{11}$ (divisible by 11)

and $a_{n}=2\cdot{4^{2n+1}}+3^{3n+1}$ is divisible by 11

It follows that $a_{n+1}=2\cdot{4^{2n+3}}+3^{3n+4}$ is also divisible by 11