Answer
See below.
Work Step by Step
1. Let $P(n)$ be the statement to be proved.
2. Test for $n=1,2,3$, we have $g_1=2^1+1=3, g_2=2^2+1=5, g_3=3g_2-2g_1=3(5)-2(3)=9$ while with the formula $g_3=2^63+1=9$, thus $ P(1),P(2),P(3)$ are all true.
3. Suppose it is true for $n\le p, (p\gt3)$, that is $g_p=2^p+1, g_{p-1}=2^{p-1}+1, ...$
4. For $n=p+1$, we have $g_{p+1}=3g_p-2g_{p-1}=3(2^p+1)-2(2^{p-1}+1)=3\cdot2^p+3-2^p-2=2\cdot2^p+1=2^{p+1}+1$ confirming the formula.
5. Thus $P(p+1)$ will also be true and we have proved the statement using mathematical induction.