# Showing that $a^n-1 \geq n\left(a^{\frac{n+1}{2}}-a^\frac{n-1}{2}\right),a>1$

How can I show that:

$$a^n-1 \geq n\left(a^{\frac{n+1}{2}}-a^\frac{n-1}{2}\right)$$

$$\sum_{k=0}^{n-1} a^k \geq na^\frac{n-1}{2}$$

$$a>1, n\in\mathbb{N}$$

without studying the function

$$f(x)=x^n-1 - n\left(x^{\frac{n+1}{2}}-x^\frac{n-1}{2}\right)$$?

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$$a^n-1=(a-1)(a^{n-1}+a^{n-2}+..+a+1)$$

By AM-GM

$$a^{n-1}+a^{n-2}+..+a+1 > na^{\frac{n-1}{2}}$$

Thus

$$a^n-1 >(a-1)na^{\frac{n-1}{2}}$$

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Thanks for noting me that. I misread the title. :) – Babak S. Aug 28 '12 at 13:17
Thank you for your answer! – Chon Aug 28 '12 at 16:19