Problem 33 (difficulty: 3/10)

Prove that

\(\displaystyle \left(1-\frac14\right) \left(1-{1\over 9}\right) \ldots \left(1-{1\over n^2}\right) = {n+1\over 2n} \)

Solution:

Induction: The statement is true for \(\displaystyle n=1\), and

\(\displaystyle a_{n+1}=\left(1-\frac1{(n+1)^2}\right)a_n,\)

assuming that the statement is true fo \(\displaystyle a_n\) we get

\(\displaystyle a_{n+1}=\left(1-\frac1{(n+1)^2}\right)\frac{n+1}{2n}=\frac{n+2}{2n+2}.\)

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