Prove that `((n + 1)/(2))^(n) gt n!`

Statement-1: If A = (300)^( 600) , B =600!, C= (200)^(600) , then A gt B gt C . Statement-2: ((n)/(2) )^(n) gt n! gt ((n)/(3) )^(n) for n gt 6 .

Prove that [(n+1)//2]^n >(n !)dot

Prove that [(n+1)//2]^n >(n !)dot

Prove that , (1)/(n+1) + (1)/(n+2) + ………….+ (1)/(2n) gt 13/24 for all natural numbers n > 1

Prove that 3^(n+1)gt3(n+1)

Using the principle of mathematical induction. Prove that (1^(2)+2^(2)+…+n^(2)) gt n^(3)/3 " for all values of " n in N .

Use Mathematical induction to prove that (1 + x)^(n) gt 1 + nx for n ge 2, x gt - 1, x ne 0

Use principle of mathematical induction, to prove that (1 + x) ^(n) gt 1 + nx , " for " n ge 2 and x gt = -1, (ne 0 )

Prove that n! (n+2)=n!+(n+1)! .

Prove that ((2n)!)/(2^(2n)(n!)^(2))<=(1)/(sqrt(3n+1)) for all n in N