Question

(1-1/2²)(1-1/3²)......(1-1/n²)=(n+1)/2n, for all natural numbers, n ≥ 2.

Answer 1

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

for all natural numbers, n ≥ 2

We, observe that P (2) is true, since

Assume that P(n) is true for some k ∈ N, i.e.,

Now, to prove that P (k + 1) is true, we have

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, n ≥ 2.