Discussion
What is the value of \( (1 - \frac{1}{3})(1 - \frac{1}{4})(1 - \frac{1}{5}) \cdots (1 - \frac{1}{10}) \)?
Answer: C
Correct answer: \( \frac{1}{5} \)
Write each factor as a single fraction: \[ 1 - \frac{1}{n} = \frac{n-1}{n}. \] The product becomes \[ \frac{2}{3} \times \frac{3}{4} \times \frac{4}{5} \times \cdots \times \frac{9}{10}. \] Every numerator cancels against the denominator of the previous factor - a telescoping product - leaving only the first numerator and the last denominator: \[ \frac{2}{10} = \frac{1}{5}. \] The general result is that \[ \prod_{n=k}^{m}\left(1 - \frac{1}{n}\right) = \frac{k-1}{m}, \] which here is \( 2/10 \). Recognising the cancellation is the whole of the work; multiplying the eight fractions out is slow and prone to error. The same structure appears with \( \left(1 + \frac{1}{n}\right) \) factors, which give \( \frac{m+1}{k} \) instead - the fraction inverts, so it is worth checking which form is presented.
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