Prove $5^n + 2 \cdot 3^n - 3$ is divisible by 8 $\forall n\in \mathbb{N}$ (using induction) 
Prove $5^n + 2  \cdot 3^n - 3$ is divisible by 8 $\forall n\in \mathbb{N}$

Base case $n = 1\to 5 + 6 - 3 = 8 \to 8 \mid 8 $
Assume that for some $n \in \mathbb{N}\to 8 \mid 5^n + 2  \cdot 3^n - 3$
Showing $8 \mid 5^{n+1} + 2  \cdot 3^{n+1} - 3$
$$5^{n+1} + 2  \cdot 3^{n+1} - 3$$ $$5\cdot 5^n + 2\cdot 3\cdot 3^n - 3$$ $$ (5^n + 2\cdot 3^n - 3) + 4\cdot 5^n + 2\cdot 2\cdot3^n $$ $$ 5\cdot(5^n + 2\cdot 3^n - 3) + 4\cdot 5^n + 2\cdot 2\cdot3^n - 4\cdot(5^n + 2\cdot 3^n - 3)$$
$$ [5\cdot(5^n + 2\cdot 3^n - 3)]  - [4\cdot 3^n - 12]$$
The first term divides by 8 but I am not sure how to get the second term to divide by 8.

$$4 \cdot 5^n+4 \cdot 3^n$$ is clearly divisible by 8 cause once you take the 4 out it is the sum of two odd numbers.
