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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.

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1 Answer 1

up vote 19 down vote accepted

Hint:$1+2+2^2+\dots 2^{n-1}=2^n-1$



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

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Again you beat me :) (+1) –  O.L. Jun 19 '13 at 10:45
Really???@O.L. Thank by the way... –  Abhra Abir Kundu Jun 19 '13 at 10:46
It almost hurts for its simplicity. +1 –  DonAntonio Jun 19 '13 at 10:49
Thanks @DonAntonio –  Abhra Abir Kundu Jun 19 '13 at 10:50
Thank you for helping me. This is the best answer for this question. –  A.D Jun 19 '13 at 10:56

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