Prove that `(n!)^(2) lt n^(n) n! lt (2n)!` for all positive integers n.

Prove that (n !)^2 < n^n n! < (2n)! , for all positive integers n.

Prove that (n !)^2 < n^n(n !)<(2n)! for all positive integers n

Prove that 2^n > n for all positive integers n.

Using mathematical induction prove that d/(dx)(x^n)=n x^(n-1) for all positive integers n.

Using binomial theorem, prove that 5^(4n)+52n-1 is divisible by 676 for all positive integers n.

Let P(n) :n^(2) lt 2^(n), n gt 1 , then the smallest positive integer for which P(n) is true? (i) 2 (ii) 3 (iii) 4 (iv) 5

Using permutation or otherwise, prove that (n^2)!/(n!)^n is an integer, where n is a positive integer. (JEE-2004]

Let P(n): (2n+1) lt 2^(n) , then the smallest positive integer for which P(n) is true?

Prove that 2 le (1+ (1)/(n))^(n) lt 3 for all n in N .

Given that u_(n+1)=3u_n-2u_(n-1), and u_0=2 ,u_(1)=3 , then prove that u_n=2^(n)+1 for all positive integer of n