If `A=(""_(1)^(3)" "_(-1)^(-4))`,prove by induction that, `A^(n)=(""_(n)^(1+2n)" "_(1-2n)^(-4n))` where n is a positive integer.

Prove by induction method that n(n^(2)-1) is divisible by 24 when n is an odd 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.

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

P(n):11^(n+2)+1^(2n+1) where n is a positive integer p(n) is divisible by -

If A={:[(a,b),(0,1)]:}, prove by mathematical induction that, A^(n)= [{:(,a^(n), (b(a^(n) -1))/(a-1)),(,0,1)]:} for every positive integer n.

If A={:[(1,1,1),(0,1,1),(0,0,1)] , Prove by mathematical induction that, A^(n)={:[(1,n,(n(n+1))/2),(0,1,n),(0,0,1)] for every positive integer n.

Prove that (""^n C_0)/x-(""^n C_1)/(x+1)+(""^n C_2)/(x+2)-.....+(-1)^n(""^n C_n)/(x+n)=(n !)/(x(x+1) . . . (x-n)), where n is any positive integer and x is not a negative integer.

Prove that sum_(k=1)^(n-1)(n-k)cos(2kpi)/n=-n/2 , where ngeq3 is an integer

By mathematical induction prove that, (2^(2n)-1) is divisible by 3 where nge1 is an integer.