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

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

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

Prove that 6^(2n)-35n-1 is divisible by 1225 for all natural numbers of n.

The value of natural number 'a' for which sum_(k=1)^(n)f(a+k)=16(2^(n)-1) , where the function satisfies the relation f(x+y)=f(x).f(y) for all natural numbers x, y and further f(1)=2 is

Statement -1 : For every natural number n ge 2, (1)/( sqrt1) + (1)/( sqrt2) + (1)/( sqrt3) + … + (1)/( sqrtn) gt sqrtn . Statement - 2 : For every natural number n ge 2, sqrt(n(n+1) ) lt n+1

What is the H.C.F of n and n+1 where n is a natural number?

Statement -1: For each natural number n, (n+1)^(7) - n^(7) -1 is divisible by 7 . Statement - 2 : For each natural number n, n^(7) - n is divisible by 7.