Prove that `2^n >1+nsqrt(2^(n-1)),AAn >2` where `n` is a positive integer.

Solve (x-1)^(n)=x^(n), where n is a positive integer.

Using permutation or otherwise, prove that (n^2)!/(n!)^n is an integer, where n is a positive integer. (JEE-2004]

What are the limits of (2^(n)(n+1)^(n))/(n^(n)) , where n is a positive integer ?

If A,=[[3,-41,1]], then prove that A^(n),=[[1+2n,-4n][2n], where n is any positive integer.

The sum of the digits in (10^(2n^(2)+5n+1)+1)^(2) (where n is a positive integer),is

The equatin 2x = (2n +1)pi (1 - cos x) , (where n is a positive integer)

Prove that (a)(1+i)^(n)+(1-i)^(n)=2^((n+2)/(2))*cos((n pi)/(4)) where n is a positive integer. (b) (1+i sqrt(3))^(n)+(1-i sqrt(3)^(n)=2^(n+1)cos((n pi)/(3)), where n is a positive integer