prove the inequality using inequalities like AM GM HM OR CAUCHY or WEIRSTRASS ETC.

Question

The inequality to be proven is $$ 2^n \gt 1 + n\cdot \sqrt{2^{n-1}} for\ all\ n>2 $$ using any inequalities like am gm hm cauchy schwarz tchebychev etc I recently studied inequalities came across this question please help

Answer 1

If n $\epsilon \mathbb{N}$, apply AM-GM on $1, 2, 2^{2}......., 2^{n-1}$. $$\frac{1 + 2 + 2^{2} +......+2^{n-1}}{n} \geqslant \sqrt[n][1*2*....*2^{n-1}]$$ $$\implies \frac{2^{n} - 1}{n} \geqslant \sqrt[n][2^{n*(n-1)/2}]$$ $$\implies \frac{2^{n} - 1}{n} \geqslant \sqrt[2][2^{n-1}]$$ $$\implies 2^{n} \geqslant 1 + n*\sqrt[2][2^{n-1}]$$ Equality holds at n=1.