1.3 +2.3² + 3.3³ +….+n.3ⁿ =

Let P(n) = 1.3 +2.3² + 3.3³ +….+n.3ⁿ =
P(1) is true.
Let P(k) be true
P(k) = 1.3 +2.3² + 3.3³ +….+3.3^k =
To prove P( k +1) is true using P(k)
P(k +1) = 1.3 +2.3² + 3.3³ +….+3.3^k + 3.3^k+1
              =
              =
              = ………….. (1)
L.H.s of ( 1)
1.3 + 2.3² + 3.3³ +………k.3^k + (k+1) 3^K+1.

              = +(k+1) 3^k+1.
              =
              =
              =
              =
              =
            = R.H.S of (1)

P(K+1) is true.

By the Principle of mathematical induction, P(n) is true for all values of n where n N

Hence proved.