Prove that `underset(xrarrinfty)Lt(1^3+2^3+…..+n^3)/((n-1)^4)=1/4`, where n is a+ve integer.

Show that (-sqrt(-1))^(4n+3)=i , where n is a positive integer.

underset(nrarrinfty)lim (1^2+2^2+3^2+………+n^2)/n^3 is equal to:

Prove that underset(r=1)overset(k)sum(-3)^(r-1) .^(3n)C_(2r-1) = 0 , where k = (3n)/2 and n is an even integer.

Show that the middle term in the expansion of (1+x)^(2n) is (1.3.5….(2n-1))/(n!) 2^(n)x^(n) , where n is a positive integer.

If A= ((-1,-4),(1,3)) , then prove by Mathematical Induction that : A^n = ((1-2n,-4n),(n,1+2n)) , where n in N

If A= ((3,-4),(1,-1)) , then prove by Mathematical Induction that : A^n = ((1+2n,-4n),(n,1-2n)) , where n in N

Prove the following by principle of mathematical induction : (a) If A=[{:(11,-25),(4,-9):}] , then A^(n)=[{:(1+10n,-25n),(4n,1-10n):}] where n is a positive integer .

Prove by mathematical induction that 1+3+3^2+…………+3^(n-1)=(3^n -1)/2