Question

Prove that 2^n >1+n√[2^(n-1)] for n>2, using the Inequalities

Answer 1

$2^n >1+n\sqrt{2^{n-1}}$ for $n>2$

$2^n-1-n\sqrt{2^{n-1}}>0$

taking derivative of the left part

$2^n*ln(2)-(n*\sqrt{2^{n-1}}*ln(2))/2-\sqrt{2^{n-1}}$

by looking up/making a graph of this function you can see that for n>2 its always positive. It means that that left part will always grow for n>2, so now we only need to check for the inequality to be true for n=2,

$2^2-1-2\sqrt{2^{2-1}}>0$

$3-2*\sqrt2>0$

$\sqrt2$ is approximately 1.414 so the inequality is true for n=2 and any n>2