The value of $\left(1 - \frac{1}{2}\right)\left(1 - \frac{1}{3}\right)\left(1 - \frac{1}{4}\right)\cdots\left(1 - \frac{1}{m}\right)$ is

### Concept & Formula This problem involves a telescoping product where successive terms simplify and lead to widespread cancellation between the numerators and denominators. ### Step-by-Step Solution Simplify each individual factor in the product: $$1 - \frac{1}{2} = \frac{1}{2}$$ $$1 - \frac{1}{3} = \frac{2}{3}$$ $$1 - \frac{1}{4} = \frac{3}{4}$$ $$\vdots$$ $$1 - \frac{1}{m} = \frac{m - 1}{m}$$ Substitute these simplified fractions back into the product: $$\left(\frac{1}{2}\right) \times \left(\frac{2}{3}\right) \times \left(\frac{3}{4}\right) \times \cdots \times \left(\frac{m-2}{m-1}\right) \times \left(\frac{m-1}{m}\right)$$ Notice the cancellation pattern: the denominator of each fraction cancels out with the numerator of the next fraction. The only terms remaining after this cascading cancellation are the first numerator and the last denominator: $$\frac{1}{m}$$ ### Exam Strategy & Shortcut For telescoping products of the type $\left(1 - \frac{1}{k}\right)$, substitute a small value for $m$ to find the pattern. Let $m = 3$: $$\left(1 - \frac{1}{2}\right)\left(1 - \frac{1}{3}\right) = \frac{1}{2} \times \frac{2}{3} = \frac{1}{3}$$ Since substituting $m = 3$ yields $\frac{1}{3}$, the general formula must be $\frac{1}{m}$. ### Common Pitfall Students sometimes write the remaining terms in reverse as $\frac{m}{1}$, forgetting that the cancellation pattern eliminates the final numerator, leaving the denominator intact. ### Final Answer **Therefore, the correct answer is 1/m.**