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

Prove that : If n is a positive integer and x is any nonzero real number, then prove that C_(0)+C_(1)(x)/(2)+C_(2).(x^(2))/(3)+C_(3).(x^(3))/(4)+….+C_(n).(x^(n))/(n+1)=((1+x)^(n+1)-1)/((n+1)x)

Prove that : If n is a positive integer then prove that i) C_(0)+C_(1)+C_(2)+……+C_(n)=2^(n)

Prove that : If n is a positive integer, then prove that C_(0)+(C_(1))/(2)+(C_(2))/(3)+….+(C_(n))/(n+1)=(2^(n+1)-1)/(n+1).

If A = [(cos theta, sin theta),(-sin theta, cos theta)] then show that for all the positive integers n, A^(n) = [(cos n theta,sin n theta),(-sin n theta,cos n theta)] .

Using binomial theorem, prove that 50^(n)-49n-1 is divisible by 49^(2) for all positive integers n.

Prove that x^n-y^n is divisible by x - y for all positive integers n.

The greatest positive integer which divides (n+16) (n+17) (n+18)(n+19), for all positive integers n is

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