If `A=[3-4 1-1]` , then prove that `A^n=[1+2n-4nn1-2n]` , where n is any positive integer.

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

((1+i)/(1-i))^(4n+1) where n is a positive integer.

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

Prove the following by principle of mathematical induction If A=[[3,-4] , [1,-1]] , then A^n=[[1+2n, -4n] , [n, 1-2n]] for every positive integer n

If A=[1101], prove that A^(n)=[1n01] for all positive integers n.

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

If A=[[1,1],[1,1]] ,prove that A^n=[[2^(n-1),2^(n-1)],[2^(n-1),2^(n-1)]], for all positive integers n.

If ((1+i)/(1-i))^x=1 then (A) x=2n+1 , where n is any positive ineger (B) x=4n , where n is any positive integer (C) x=2n where n is any positive integer (D) x=4n+1 where n is any positive integer

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