The series doesn't look right. The nth term would be n.3^2 not n.3^n. Alternatively, the series would be:
(1)(3)+(2)(3^2)+(3)(3^3)+...+(n)(3^n).
Let's assume it's the first alternative: 3^2(1+2+3+...+n)=9n(n+1)/2. Since this doesn't give us the format expressed in the equation, we have to assume the second alternative:
∑(n)(3^n) for numbers 1, 2, 3, ..., n.
S[n+1]=S[n]+(n+1)3^(n+1), where S[n] means the sum to n terms.
Let's suppose S[n]=(3/4)((2n-1)3^n+1), then:
S[n+1]=(3/4)((2n-1)3^n+1)+(n+1)3^(n+1)=
(1/4)((2n-1)3^(n+1)+3+4(n+1)3^(n+1)=(1/4)((6n+3)3^(n+1)+3)=
(3/4)((2n+1)3^(n+1)+1)=(3/4)((2(n+1)-1)3^(n+1)+1)=S[n+1] according to the supposed formula.
Also, the base case for n=1, S1=(3/4)(3+1)=3, the first term (1)(3).
So the formula works for consecutive values of n, and it's correct for S1, therefore, by induction, the formula is correct for the general sum up to n.