3^(n)>2^(n)

lim_(n rarr oo)(3^(n+1)+2^(n+2))/(3^(n-1)+2^(n-2)) =

1.3+2.3^(2)+3.3^(3)+............+n.3^(n)=((2n-1)3^(n+1)+3 )/(4)

underset(n to oo)lim(((n+1)(n+2)...3n)/(n^(2n)))^((1)/(n)) is equal to

lim_(n rarr infty ) [((n+1)(n+2)...3n)/(n^(2n))]^(1//n) is equal to

lim_(ntooo)(((n+1)(n+2)....3n)/(n^(2n)))^(1//n) is equal to

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

lim_(n to infty)(1^(2)/(1-n^(3))+2^(2)/(1-n^(3))+…+n^(2)/(1-n^(3))) is equal to :

Using the principle of mathematical induction prove that : the 1.3+2.3^(2)+3.3^(3)++n.3^(n)=((2n-1)3^(n+1)+3)/(4) for all n in N.