Induction proof that $4^n > 3^n+2^n$ for $n\ge2$

Question

This is a problem with induction and proofs but I'm not sure how to start with proving this one.

$$\text{Show that for any $n \geq 2$, $4^n > 3^n+2^n$}$$

Answer 1

Note that $$4^{n+1} = \underbrace{4 \cdot 4^n > 4 (3^n+2^n)}_{\text{From induction hypothesis}} = \overbrace{4 \cdot 3^n + 4 \cdot 2^n > 3 \cdot 3^n + 2 \cdot 2^n}^{\text{Since }4 > 3 \text{ and }4>2} = 3^{n+1} + 2^{n+1}$$ and you are done.