General solution to homogeneous difference equation

Question

With a given example

$$ a_{n-1} = ca_{n-2} $$

general solution:

$$ a_{n} = c . c . a_{n-2} $$
$$ = c . c . a_{n-3} $$
$$ = c^n a_0 $$

Question: Find the general solution for the homogeneous equation $$ a_{n} = 5a_{n-1} $$

General solution:

$$ a_{n} = 5^n a_{0} $$

Is my general solution correct based on the given example?

Answer 1

Even if that first equation is a typo, that is the correct general solution. If a proof is necessary, I suggest using Induction on $n$.