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

Given that u_(n+1)=3u_n-2u_(n-1), and u_0=2 ,u_(1)=3 , then prove that u_n=2^(n)+1 for all positive integer of n

Using mathematical induction prove that (d)/(dx) (x^(n))= n x^(n-1) for all positive integers n.

Prove by induction that if n is a positive integer not divisible by 3. then 3^(2n)+3^(n)+1 is divisible by 13.

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

Let an denote the number of all n-digit positive integers formed by the digits 0, 1 or both such that no consecutive digits in them are 0. Let b_n = the number of such n-digit integers ending with digit 1 and c_n = the number of such n-digit integers ending with digit 0. The value of b_6 , is

If a and b are distinct integers, prove that a-b is a factor of a^n - b^n , whenever n is a positive integer.

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

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

Find the sum of : the first n positive integers

If the arithmetic mean of the product of all pairs of positive integers whose sum is n is A_n then lim_(n->oo) n^2/(A_n) equals to