Prove the following by the principle of mathematical induction for
every natural number n.1.2.3 + 2.3.4 + 3.4.5 + ….. + n(n + 1)(n + 2) = n(n + 1)(n +2)(n +3)/4.

Let P(n) : 1.2.3 + 2.3.4 + 3.4.5 + ….. + n(n + 1)(n + 2) = n(n + 1)(n +2)(n + 3)/4

P(1) : 1.2.3 = 1(1 + 1)(1 +2)(1 + 3)/4 = 1.2.3.4/4 = 6 which is true. Thus P(n) is true for n = 1.
Assume P(k) : 1.2.3 + 2.3.4 + 3.4.5 + ….. + k(k + 1)(k + 2) = k(k + 1)(k +2)(k + 3)/4
To prove: P(k + 1) : 1.2.3 + 2.3.4 + 3.4.5 + ….. + (k + 1)(k + 2)(k + 3) = (1/4)(k + 1)(k + 2)(k + 3)(k + 4)
L.H.S = 1.2.3 + 2.3.4 + 3.4.5 + ….. + (k + 1)(k + 2)(k + 3)
        = 1.2.3 + 2.3.4 + 3.4.5 + ….. + k(k + 1)(k + 2) + (k + 1)(k + 2)(k + 3)
        = k(k + 1)(k +2)(k + 3)/4 + (k + 1)(k + 2)(k + 3)
        = (k + 1)(k + 2)(k + 3)(k + 4)/4
        = (1/4)(k + 1)(k + 2)(k + 3)(k + 4)
        = 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 for all natural numbers.