Use Principle of Mathematical Induction to show that for all $n\:\in \mathbb{N}$, $a_n=2^{n+2}\cdot 5^{2n+1}+3^{n+2}\cdot 2^{2n+1}$ is divisible by 19
You're almost there! Hint for the finish: $$ 2^{k+2}\cdot 5^{2k+1}\cdot 50+3^{k+2}\cdot 2^{2k+1}\cdot 12\\ = 12\left(2^{k+2}\cdot 5^{2k+1}+3^{k+2}\cdot 2^{2k+1}\right) + 38\left(2^{k+2}\cdot 5^{2k+1}\right) $$