Prove the following by the principle of mathematical induction:

1.3 + 2.4 + 3.5 + … + n . (n + 2)

Let P(n): 1.3 + 2.4 + 3.5 + … + n.(n + 2) =

For n = 1

P(1): 1.3 = .1.(2)(9)

= 3 = 3

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

Now,

For n = k

= P(n): 1.3 + 2.4 + 3.5 + … + k . (k + 2)= - - - - - (1)

We have to show that

= 1.3 + 2.4 + 3.5 + … + k . (k + 2) + (k + 3) =

Now,

= {1.3 + 2.4 + 3.5 + … + k (k + 2)} + (k + 1)(k + 3)

= k(k + 1)(2k + 7) + (k + 1)(k + 3) using equation (1)

= (k + 1)

= (k + 1)

= (k + 1)

= (k + 1)

= (k + 1)

= (k + 1)

= (k + 1)(k + 2)(2k + 9)

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

Hence, P(n) is true for all n∈ N