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

Prove each of the statements by the principle of mathematical induction : sqrtn lt (1)/(sqrt1) + (1)/(sqrt2) + (1)/(sqrt3) + …….+ (1)/(sqrtn) , for all natural numbers n gt= 2

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

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

Prove that (n!) (n + 2) = [n! + (n + 1)!]

Prove each of the statements by the principle of mathematical induction : 1+2+2^n + ….. + 2^n = 2^(n+1) - 1 for all natural numbers n .

prove using mathematical induction, n(n+1)(n+5) is divisible by 6 for all natural numbers

Prove the following by using the principle of mathematical induction for all n in N (1)/(n+1) + (1)/(n+2) +…..1/(2n) gt 13/24, n gt 1

Prove that .^(n-1)C_(3)+.^(n-1)C_(4) gt .^(n)C_(3) if n gt 7 .

Prove that, 1^2 + 2^2 + …..+ n^2 gt (n^3)/(3) , n in N

Use the principle of mathematical induction to show that 5^(2n+1)+3^(n+2).2^(n-1) divisible by 19 for all natural numbers n.