# How do you prove this using mathematical induction

I am unsure if my proof method is correct for this problem here:

Use mathematical induction to prove that for each integer $$n ≥ 3, 4^n ≤ 5^n − 60.$$

This is my working out so far:

Base case $$n = 3,$$ $$4^3 ≤ 5^3 - 60$$. P(3) is true

Inductive hypothesis: Assume P(k) is true. Show $$P(k+1)$$ is true.

$$4^{k+1} ≤ 5^{k+1} - 60$$.

LHS = $$4^{k+1}$$

= $$4(4^k)$$

$$<4(5^k)-60$$

$$<5(5^k)-60$$

= $$5^{k+1}-60$$

Therefore, by the principle of mathematical induction?, for every integer $$n ≥ 3, 4^n ≤ 5^n − 60.$$

Not quite sure if this is the right way to prove it.

• Note that the right hand side becomes $4 \times 5^k - 240 < 5^{k+1} - 60$. – Ertxiem - reinstate Monica Apr 6 at 16:02

$$4(4^k)$$
$$<4(5^k)-60$$
It should be $$4(4^k) <4(5^k-60) = 4\cdot 5^k-240 <5\cdot 5^k-240 <5^{k+1}-60$$