Prove the Following by Using the Principle of Mathematical Induction for All N ∈ N: (1+3/1)(1+ 5/4)(1+7/9)...(1 + ((2n + 1))/N^2) = (N + 1)^2

Question

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

(1+3/1)(1+ 5/4)(1+7/9)...`(1 + ((2n + 1))/n^2) = (n + 1)^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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