prove that `2nlt(n+2)!` for all natural numbers n.

Prove that 2n +1 lt 2^(n) for all natural numbers n ge3

prove that n^(2)lt2^(n) , for all natural number n≥5 .

prove that n^(2)lt2^(n) , for all natural number n≥5 .

Prove each of the statements by the principle of mathematical induction : 2n lt (n + 2) ! for all natural number n .

prove that 2+4+6+…2n=n^(2)+n , for all natural numbers n.

prove that 2+4+6+…2n=n^(2)+n , for all natural numbers n.

By the principle of mathematical induction prove that for all natural numbers 'n' the following statements are true, 2+2^(2)+2^(3)+…..+2^(n)=2(2^(n)-1)

Prove that (1+x)^n ge1+n x , for all natural number 'n', where x gt-1 .

For a gt 1 , prove that (1+a)^(n) ge (1+an) for all natural numbers n .

Prove that (1+x)^(n) ge (1+nx) for all natural number n where x gt -1