How can I prove that $(\dfrac{( n+1)}{2})^n$ is greater than $n!$ where $n$ is natural number greater than $1$?
35611
How can I show that $n! \leqslant \left(\frac{n+1}{2}\right)^n$?
59479
For a positive integer $n\geq 2$ with divisors $1=d_1<d_2<\cdots<d_k=n$, prove that $d_1d_2+d_2d_3+\cdots+d_{k-1}d_k<n^2$
2554
Show that if $n>2$, then $(n!)^2>n^n$.
142347
What is the highest power of $n$ in $(n^r-1)!$
8609
Prove by induction that $n!>2^n$
3043
Prove $n!>2^n$ for $n\geq4$ using induction.
12967
Find all $n>1$ such that $\dfrac{2^n+1}{n^2}$ is an integer.
97684
Show $\frac{(2n)!}{n!\cdot 2^n}$ is an integer for $n$ greater than or equal to $0$
22073
Prove by induction that $n^2<n!$
149758
How to prove $\log n < n$?
20691
prove that $\frac{(2n)!}{(n!)^2}$ is even if $n$ is a positive integer
80216
Prove if $n \in \mathbb N$, then $\frac{1}{2!}+\cdots+\frac{n}{(n+1)!}=1-\frac{1}{(n+1)!}$
121694
How to prove $\log n \leq \sqrt n$ over natural numbers?
34767
Proving $\left(1+\dfrac{1}{1^3}\right)\left(1+\dfrac{1}{2^3}\right)\cdots\left(1+\dfrac{1}{n^3}\right)<3$ for all positive integers $n$
10629
How to prove $4(n!)>2^{n+2}$ for $ n\geq 4$ with induction
102334
How can you prove $\frac{n(n+1)(2n+1)}{6}+(n+1)^2= \frac{(n+1)(n+2)(2n+3)}{6}$ without much effort?
145891
Prove that $n! > \sqrt{n^n}, n \geq 3$