Une inégalité sous contraintes

Soit
$n \in \mathbb{N}$ et soient$a_2, \ldots, a_n$ des réels strictement positifs tels que$\prod_{i=2}^n a_i = 1$ .Montrer que:
$\prod_{i=2}^n (1+a_i)^i$ >$n^n$

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That's what I call a brief and perfect answer!

@MouchtakiOmar thank you for the compliment !Can you post your solution if you don't mind of course!?

I'm afraid, I've just noticed that I don't completely agree with your AMGM inequality!
Can you give me detail on how you used it?

Sorry,my proof isn't true.I will fix it tomorrow.!

No problem! I'm sorry, I read it too fast!

I can't wait until tomorrow hhh![alt text]( image url)

hahahha!
Why do you start with a sum when the exercise is a product?
I think you almost found it, just write it again with a product!

hhh I really want to write a product,but my mind had gone with the sum!for the third time,I will fix it again!