Question

For all n ∈ N, by using principle of mathematical induction, then prove that:

1.2.3 + 2.3.4 + …… + n(n + 1) (n + 2) = (n(n + 1)(n + 2)(n + 3))/4

Answer 1

Let given statement is P(x) where n ∈ N. i.e.,

P(n) = 1.2.3 + 2.3.4 +………+ n(n + 1)(n + 2)

Hence, given statement is true for n = 1,

i.e, P(1) is true

Let given statement is true for n = m, 

i.e, P(m) is true

Hence, the given statement is also true for n = m + 1, 

i.e, P(m + 1) is also true.

Hence, by the principle of mathematical induction.

We can say that the given statement is true for each natural numbers n ∈ N.

Hence Proved.