Answer
See below.
Work Step by Step
1. Let $P(n)$ be the statement to be proved.
2. Test for $n=0,1,2$, we have $f_0=3(2^0)+2(5^0)=5, f_1=3(2^1)+2(5^1)=16, f_2=7f_1-10f_0=7(16)-10(5)=62$ while with the formula $f_2=3(2^2)+2(5^2)=62$, thus $ P(0),P(1),P(2)$ are all true.
3. Suppose it is true for $n\le p, (p\gt2)$, that is $f_p=3(2^p)+2(5^p), e_{p-1}=3(2^{p-1})+2(5^{p-1})...$
4. For $n=p+1$, we have $f_{p+1}=7f_p-10f_{p-1}=7(3(2^p)+2(5^p))-10(3(2^{p-1})+2(5^{p-1}))=21(2^{p})+14(5^{p})-30(2^{p-1})-20(5^{p-1})=6(2^{p})+10(5^{p})=3(2^{p+1})+2(5^{p+1})$ confirming the formula.
5. Thus $P(p+1)$ will also be true and we have proved the statement using mathematical induction.