Show that inequality is correct for natural $n$

Question

Show that the following inequality is correct for all natural $n$ :

$$(2n+1)^n\geq(2n)^n+(2n-1)^n$$

I've tried throwing the $(2n-1)^n$ or $(2n)^n$ on the left side and using formula of subtraction of nth powers, but I can't prove it. Any hints or ideas will be appreciated.

Answer 1

Divide across by $(2n)^n$ to get the equivalent $(1+{1 \over 2n})^n - (1-{1 \over 2n})^n \ge 1$.

Using the binomial expansion we have $\sum_{k=0}^n \binom{n}{k} ({1 \over 2n})^k (1-(-1)^k) \ge \binom{n}{1}({1 \over 2n})^1 (1-(-1)^1) = 1$.