I would use induction here. I will assume
1. Show the base case is true:
True.
2. State the induction hypothesis :
Next, look at:
and
That is, see if gives you a true inequality.
What do you find?
Our teacher asked this question in a homework :
Prove :
I found these pages which gave me some clues :
number theory - prove that $(2n)!/(n!)^2$ is even if $n$ is a positive integer - Mathematics
Prove that 2n choose n is less than or equal to 4^n? - Yahoo! Answers
However, I can't figure that out ! According to the 2nd page, if the right member was only , then the left member would be less than or equal to the right member.
I just don't know what to do.
You help would be appreciated.
Thanks.
I would use induction here. I will assume
1. Show the base case is true:
True.
2. State the induction hypothesis :
Next, look at:
and
That is, see if gives you a true inequality.
What do you find?
The PDF file is what I got... but I'm not too sure how to compare left with right...
I PROBABLY MADE ALGEBRAIC ERRORS !
I know that :
-> is
-> the left "big parenthesis" is > the right "big parenthesis"
But what about the ?!
I'm surely this is much simpler than I think, but my brain's kind of burnt right now !
Thanks !
The PDF is in French, but I wrote English translation in red.
That is essentially the same technique I have in mind, but I can get you there much more simply. Unfortunately, I don't have the time at the moment. I will be glad to elaborate more in a few hours. If you compute the differences I suggest, it should all become clear.
You should verify that:
and
So, next, let's verify:
This is obviously true for . So, multiplying the induction hypothesis by:
and then adding the result to , we will obtain , thereby completing the proof by induction.