Show that $2^n>1+n\sqrt{2^{n-1}}$
Solution 1:
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 $2^n-1=\displaystyle \sum_{k=0}^{n-1}2^{k}\ge n(2^{(\sum_{k=0}^{n-1}k)})^{\frac{1}{n}}=n2^{\frac{n-1}{2}}$