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.
Best Answer
This is wrong:
$4(4^k)$
$<4(5^k)-60$
It should be $$4(4^k) <4(5^k-60) = 4\cdot 5^k-240 <5\cdot 5^k-240 <5^{k+1}-60$$