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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.

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.