3.6 +6.9+9.12+....+3n(3n+3) = 3n(n+1) (n+2)

= 3*2*3 = 18

therefore P(1) is true

for n = k. P(k) = 3k(k+1)(k+2) assume p(k) to be true.

P(k+1) = p(k) + (k+1)^{th }term

3k(k+1)(k+2) + 3(k+1)(3k+4)

3(k+1) [ k(k+2) + 3k+4 ]

3(k+1) [ k2+5k+4]

3 ( k+1 ) ( k+2 ) ( k+3) = RHS

Thus by the principle of mathematical induction, the statement P (*n*) is true for all .