Let P(n): 1.2 + 2.3 + 3.4 + … + n(n + 1)=

For n = 1

P(1): 1(1 + 1)=

= 1x2 =

= 2 = 2

Since, P(n) is true for n = 1

Let P(n) is true for n = k

= P(k): 1.2 + 2.3 + 3.4 + … + k(k + 1)= - - - - - (1)

We have to show that,

= 1.2 + 2.3 + 3.4 + … + k(k + 1) + (k + 1)(k + 2)=

Now,

{1.2 + 2.3 + 3.4 + … + k(k + 1)} + (k + 1)(k + 2)

=

= (k + 2)(k + 1)

=

Therefore, P(n) is true for n = k + 1

Hence, P(n) is true for all n ∈ N