Let the given statement be P(n), i.e.,
P (n) : 1 . 2 + 2 . 3 + 3 . 4 + ... + n . (n + 1) = [

n(n+1)(n+2)

2

]
First we prove that the statement is true for n = 1.
P (1) : 1 . 2 =

1(1+1)(1+2)

3

⇒ 2 =

2.3

3

= 2 , which is true
Assume P(k) is true for some positive integer k, i.e.,
1 . 2 + 2 . 3 + 3 . 4 + ... + k . (k + 1) = [

k(k+1)(k+2)

2

] ... (i)
We shall now prove that P(k + 1) is also true.
For this we have to prove that
1 . 2 + 2 . 3 + 3 . 4 + ... + k . (k + 1) (k + 2) =

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

3

L.H.S. = 1 · 2 + 2 · 3 + 3 · 4 + ... + k(k + 1) + (k + 1)(k + 2)
=

k(k+1)(k+2)

3

+ (k + 1) (k + 2) [From (i)]
= (k + 1) (k + 2) [

k

3

+1] =

(K+1)(k+2)(k+3)

3

= R.H.S.
Thus P(k + 1) is true, whenever P(k) is true.
Hence, by the principle of mathematical induction P(n) is true ∀ n ∈ N.