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

Prove that log_n(n+1)>log_(n+1)(n+2) for any natural number n > 1.

Prove that 2^(n) gt n for all positive integers n.

Prove by induction that : 2^(n) gt n for all n in N .

First term of a sequence is 1 and the (n+1)th term is obtained by adding (n+1) to the nth term for all natural numbers n, the 6th term of the sequence is

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.

Show by using the principle of mathematical induction that for all natural number n gt 2, 2^(n) gt 2n+1

If f : N xx N rarr N is such that f (m,n) = m+n , for all n in N , where N is the set of all natural numbers, then which of the following is true?

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

Prove that the function f(x)= x^(n) is continuous at x= n, where n is a positive integer.

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