Question

Prove 2^n > n for all n equal to or greater than 1. I mostly need help with how to solve the problem when it is greater than rather than equal to. Thanks for the help

Answer 1

If $n$ is an integer, you can use induction. First show the inequality holds for $n=1$. You have $2^1=2>1$, which is true.

Now assume this holds in general for $n=k$, i.e. that $2^k>k$. We want to prove the statement then must hold for $n=k+1$.

Because $2^k>k$, you have

$2^{k+1}=2\times2^k>2k$

and this must be greater than $k+1$ for the statement to be true, so we require

$2k>k+1$

for $k>1$. Well this is obviously true, because solving the inequality gives $3k>1\implies k>\dfrac13$. So you're done.

If you $n$ is any real number, you can use derivatives to show that $2^n$ increases monotonically and faster than $n$.