Prove via mathematical induction that $4n < 2^n$ for all $ n≥5$.
Solution 1:
The induction base is correct.
For the inductive step, we assume that the result holds for $n$, with $n\geq 5$; that is, are assuming that $$4n\lt 2^n,\qquad n\geq 5.$$ We want to prove that, under this assumption, $4(n+1)\lt 2^{n+1}$.
Hint the first. $4(n+1) = 4n+4 \lt 2^n+4$, with the last step using the induction hypothesis.
Hint the second. $2^{n+1} = 2\times 2^n = 2^n+2^n$.
Solution 2:
Hint
So you want to show for $n>4$, $$4n<2^n \implies 4n+4 < 2^n+2^n$$