Question

By the principle of mathematical induction, prove the following:

1.2 + 2.3 + 3.4 + … + n(n + 1) = `(n(n + 1)(n + 2))/3` for all n ∈ N.

Answer 1

Let P(n) denote the statement

1.2 + 2.3 + 3.4 + …… + n(n + 1) = `(n(n + 1)(n + 2))/3`

Put n = 1

LHS = 1(1 + 1) = 2

RHS = `(1(1 + 1)(1 + 2))/3 = (1(2)(3))/3` = 2

∴ P(1) is true.

Now assume that the statement be true for n = k

(i.e.,) assume P(k) be true

(i.e.,) assume 1.2 + 2.3 + 3.4 + …… + k(k + 1) = `(k(k + 1)(k + 2))/3` br true

To prove: P(k + 1) is true

(i.e.,) to prove: 1.2 + 2.3 + 3.4 + …… + k(k + 1) + (k + 1) (k + 2) = `((k + 1)(k + 2)(k + 3))/3`

Consider 1.2 + 2.3 + 3.4 + ……. + k(k + 1) + (k + 1) (k + 2)

= [1.2 + 23 + …… + k(k + 1)] + (k + 1) (k + 2)

= `(k(k + 1)(k + 2))/3` + (k + 1)(k + 2)

`= (k(k + 1)(k + 2) + 3(k + 1)(k + 2))/3`

`= ((k+1)(k+2)(k+3))/3`

∴ P(k + 1) is true.

Thus if P(k) is true, P(k + 1) is true.

By the principle of Mathematical ‘induction, P(n) is true for all n ∈ N.

1.2 + 2.3 + 3.4 + …… + n(n + 1) = `("n"("n + 1")("n + 2"))/3`