Q. 14

# Prove the following by the principle of mathematical induction:1.2 + 2.3 + 3.4 + … + n(n + 1) Let P(n): 1.2 + 2.3 + 3.4 + … + n(n + 1)= For n = 1

P(1): 1(1 + 1)= = 1x2 = = 2 = 2

Since, P(n) is true for n = 1

Let P(n) is true for n = k

= P(k): 1.2 + 2.3 + 3.4 + … + k(k + 1)= - - - - - (1)

We have to show that,

= 1.2 + 2.3 + 3.4 + … + k(k + 1) + (k + 1)(k + 2)= Now,

{1.2 + 2.3 + 3.4 + … + k(k + 1)} + (k + 1)(k + 2)

= = (k + 2)(k + 1) = Therefore, P(n) is true for n = k + 1

Hence, P(n) is true for all n N

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