Past exam of the year 1 of the computer science course of the University of Oxford / 2021 / Discrete mathematics, continuous mathematics, and probability / 2 / a / i / Solution

For the base case $n=2$, the product is $1-1/4=3/4=(2+1)/(2\cdot 2)$. Assume the identity holds for some $n\ge 2$. Multiplying by the next factor givesThis is the desired expression with $n$ replaced by $n+1$, so mathematical induction proves the identity for every $n\ge 2$. The factorization $1-i^{-2}=(i-1)(i+1)/i^2$ also explains the cancellation as a telescoping product.