To prove the following inequality of real numbers

Question

If $a_1,...a_n$ are all positive real numbers then prove that $$\left(\frac{a_1 + a_2 + \dots + a_n}{n}\right)^n \ge a_1a_2\left(\frac{a_3 + a_4 + \dots + a_n}{n-2}\right)^{n-2}.$$

I approached the problem by $AM > GM$ on $a_1,...a_n$, but I am unable to proceed any further for the AM GM inequality on $n-2$ numbers $a_3,..a_n$. Thank you for your help.

Answer 1

Use instead the AM-GM inequality on $$a_1, a_2, \underbrace{b, b, \dots, b}_{n-2 \text{ times}}$$ where $b = \frac{a_3 + a_4 + \dots + a_n}{n-2}$.