Induction proof with inequalities

Question

Consider the following claim: $$5^n > 4^n + 3^n + 2^n$$

(a) For what natural numbers is this claim true?

(b) Prove that your answer to (a) is correct using induction on n.

Answer 1

1. When $n=3$ then inequality holds.

2

• $n=3$, it holds.

• Suppose that $n=k$ it also holds.

• When $n=k+1$, then

$$5^{k+1}=5\times 5^k > 5 \times (4^k+3^k+2^k)=5\times 4^k+5\times 3^k+ 5 \times 2^k>4^{k+1} + 3^{k+1}+ 2^{k+1}$$