Induction when not dealing with Sigma notation

Question

How do you prove $4^n > 3^n + 2^n$ using induction? Base case would be when $n = 2$, $16 > 13$. Then assume $n = k$ so that $4^k > 3^k + 2^k$. Then let $n = k + 1$ so that $4^{k+1} > 3^{k+1} + 2^{k+1}$ But then what? What am I trying to match up so that this can be proven by induction?

Answer 1

If $\displaystyle4^k>3^k+2^k,$

$$4^{k+1}=4\cdot4^k>4(3^k+2^k)=4\cdot3^k+4\cdot2^k>3\cdot3^k+2\cdot2^k$$