Prove the Following by Using the Principle of Mathematical Induction for All N ∈ N: 1/1.2.3 + 1/2.3.4 + 1/3.4.5 + ...+ 1/(N(N+1)(N+2)) = (N(N+3))/(4(N+1) (N+2))

Question

Prove the following by using the principle of mathematical induction for all n ∈ N:

1/1.2.3 + 1/2.3.4 + 1/3.4.5 + ...+ `1/(n(n+1)(n+2)) = (n(n+3))/(4(n+1) (n+2))`

Solution

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

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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Principle of Mathematical Induction

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