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I'm trying to do homework problems and for the most part I've been getting the results. For this one though, I am having some trouble since its $2^n$ and I can't relate it properly:
Prove using simple induction that $n < 2^n$ for all $n \ge 1$.
So obviously, the basis step holds as $1 < 3$. Now, I assume that $n = k$ holds as well and have to prove for $n=k+1$. This is the step I am having trouble with as I cannot relate the induction hypothesis with what I want to end up with. Can anybody show me a model solution for this one? I think my trouble comes because of the $2^n $