Prove that 1.2 + 2.3 + 3.4 +..............n(n+1) = 1/3 n (n+1)(n+2)

Let the given statement be P(*n*), i.e.,

P(*n*):

For *n* = 1, we have

P(1): , which is true.

Let P(*k*) be true for some positive integer *k*, i.e.,

We shall now prove that P(*k* + 1) is true.

Consider

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

= [1.2 + 2.3 + 3.4 + … + *k*.(*k* + 1)] + (*k* + 1).(*k* + 2)

Thus, P(*k* + 1) is true whenever P(*k*) is true.

Hence, by the principle of mathematical induction, statement P(*n*) is true for all natural numbers i.e., *n*.

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