Past exam of the year 1 of the computer science course of the University of Oxford / 2023 / Discrete mathematics, probability and continuous mathematics / 2 / b / Solution

At $n=2$, the product is $1-1/4=3/4$, as required. Suppose the formula holds at $n\ge 2$. Appending the next factor giveswhich is exactly the formula at $n+1$. Thus mathematical induction gives $(n+1)/(2n)$ for every $n\ge 2$. The factorization $1-i^{-2}=(i-1)(i+1)/i^2$ also explains the cancellation in this telescoping product.