mathematical inductuon

Let S(n) be the statement
1x 3+ 2x 3^2 +3x3^3+...+n x 3^n = 3/4 ﹝1+(2n-1) 3^n﹞
When n = 1, LHS = 1x3 = 3
RHS = 3/4 (1+(2-1)3) = 3/4*4 = 3
S(1) is true.
Assume S(k) is true, i.e.
1x 3+ 2x 3^2 +3x3^3+...+k x 3^k = 3/4 ﹝1+(2k-1) 3^k﹞
When n = k+1,
LHS = 1x 3+ 2x 3^2 +3x3^3+...+k x 3^k + (k+1)x3^(k+1)
= 3/4 ﹝1+(2k-1) 3^k﹞+ (k+1)x3^(k+1)
= 3/4 + (2k-1)3^(k+1)/4 + (k+1)3^(k+1)
= 3/4 + 3^(k+1) [(2k-1)/4 + (k+1)]
= 3/4 + [3^(k+1)/4] (2k-1 + 4k+4)
= 3/4 + [3^(k+1)/4] (6k + 3)
= 3/4 [1+ (2k+1)3^(k+1)]
= 3/4 [1+ [2(k+1)-1]x3^(k+1)]
S(k+1) is true.
By M.I., S(n) is true for all positive integers