Range of the expression $n^n (\frac{n+1}{2})^{2n}$

Question

Prove that:

$n^n (\frac{n+1}{2})^{2n}$

1. Greater than or equal to $(\frac{n+1}{2})^3$

2. Greater than or equal to $(n!)^3$

Here $n\in \mathbb{N}$

How to prove this? It seems too complicated. Please suggest as how to initiate the solution?

Answer 1

Using $\bf{A.M\geq G.M}\;,$ We get

$$\frac{1^3+2^3+3^3+.........+n^3}{n}\geq \left(1^3\cdot 2^3\cdot 3^3\cdot.......n^3\right)^{\frac{1}{n}}$$

So $$\frac{n^2(n+1)^2}{4n}\geq \left(n!\right)^{\frac{1}{n}}\Rightarrow n^n\cdot \left(\frac{n+1}{2}\right)^{2n}\geq (n!)^3$$

Answer 2

Part 1.

For $n\in N$, we have: $n\geq \frac{n+1}{2}$

$n (\frac{n+1}{2})^2\geq (\frac{n+1}{2})^3$

$n^n (\frac{n+1}{2})^{2n}\geq (\frac{n+1}{2})^3$