Prove that g_n is divisible by 4(n>=1 n is integer)

Given information:
g_1=4
g_2=12
g_k=3g_k-2 - g_k-1 (k>=3)

How do I prove that g_n is divisible by 4?

We use the principle of strong induction (ie. Show that it is true for the base case. Then we assume it is true for all numbers up to n=m and show it must be true for n=m+1):

So, we know that it is true for all n up to 2, right? (4 and 12 are obviously divisible by 4).

Now assume it is true for all numbers up to n=m,

then we have g_m+1 = 3*g_m-1 - g_m

Because we know (by assumption that g_m-1 and g_m are divisible by 4, we can take out a factor of 4 and we will still have a integer, ie:

g_m+1 = 4(3*integer1- integer2).

This shows that g_m+1 is clearly divisible by 4, as it has a factor of 4 out front.

We have shown by induction that it is true for all m.