`(-i)^(4n+3)`, where n Is a positive integer.

Prove that (1+i)^n+(1-i)^n=2^((n+2)/2).cos((npi)/4) , where n is a positive integer.

Prove that (1+i)^n+(1-i)^n=2^((n+2)/2).cos((npi)/4) , where n is a positive integer.

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

If aa n db are distinct integers, prove that a-b is a factor of a^n-b^n , wherever 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]

If the first three terms in the expansion of (1 -ax)^n where n is a positive integer are 1,-4x and 7x^2 respectively then a =

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

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

Show that one and only one out of n ,n+2or ,n+4 is divisible by 3, where n is any positive integer.

Show that one and only one out of n ,n+2 or n+4 is divisible by 3, where n is any positive integer.