Proving Binomial Theorem with Greatest Term and Coefficient Relationship

[Quantum Mind]

Homework Statement

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.

Homework Equations

No idea.

The Attempt at a Solution

Don't know where to start.

 

[MathMarvel]

2n is obviously an even number, so your greatest coefficient occurs at the n^th term. Use the binomial expansion to also find your (n-1)^th and (n+1)^th terms (their coefficients will be equal), plug in your given boundaries for x, and do a little algebra to show that at one of the boundaries the (n-1)^th and n^th terms are equal and at the other boundary the n^th and (n+1)^th terms are equal. Hopefully this leads you in the right direction.

 

[Quantum Mind]

Thank You, it is clear now.