Proof of a binomial problem

[Step Ahead]

Show that if the greatest term in the expansion of (1+x)²ⁿ is also the greatest coefficient, then x lies between n/n+1 and n+1/n.

I have no idea how to go about this one. Hints please?

 

[daon2]

The greatest coefficient is \(\displaystyle { 2n \choose n}\). The greatest term depends on \(\displaystyle x\) but must be \(\displaystyle {2n \choose i}x^{2n-i}\) for some \(\displaystyle i\). From what you write, it seems you are assuming these quantities are equal.

You are also using ambiguous notation in what you are supposed to show.

 

[Step Ahead]

I'm getting it now, thanks.