TITLE: Show that $2^n>1+n\sqrt{2^{n-1}}$
QUESTION [10 upvotes]: If $n$ be a positive integer greater than $1$, then prove that $$2^n>1+n\sqrt{2^{n-1}}$$ 

I found this problem under $AM \ge GM$ chapter. Help me to solve this problem using  $AM \ge GM$. Thanks in advance.

REPLY [22 votes]: Hint:$1+2+2^2+\dots 2^{n-1}=2^n-1$
Edit:
Solution:

 Applying A.M.-G.M. on $\displaystyle \sum_{k=0}^{n-1}2^{k}$ we have $2^n-1=\displaystyle \sum_{k=0}^{n-1}2^{k}\ge n(2^{(\sum_{k=0}^{n-1}k)})^{\frac{1}{n}}=n2^{\frac{n-1}{2}}$