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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.

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

$4(4^k)$

$<4(5^k)-60$

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

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