When simplified, the product $\left(2 - \frac{1}{3}\right)\left(2 - \frac{3}{5}\right)\left(2 - \frac{5}{7}\right)\cdots\left(2 - \frac{997}{999}\right)$ is equal to

### Concept & Formula This problem contains a sequence where each term follows the pattern $\left(2 - \frac{2k-1}{2k+1}\right)$. Simplifying these fractions helps identify a telescoping behavior where terms eliminate each other diagonally. ### Step-by-Step Solution Simplify each factor step-by-step: $$2 - \frac{1}{3} = \frac{6 - 1}{3} = \frac{5}{3}$$ $$2 - \frac{3}{5} = \frac{10 - 3}{5} = \frac{7}{5}$$ $$2 - \frac{5}{7} = \frac{14 - 5}{7} = \frac{9}{7}$$ $$\vdots$$ $$2 - \frac{997}{999} = \frac{1998 - 997}{999} = \frac{1001}{999}$$ Write down the product with the newly simplified fractions: $$\left(\frac{5}{3}\right) \times \left(\frac{7}{5}\right) \times \left(\frac{9}{7}\right) \times \cdots \times \left(\frac{1001}{999}\right)$$ Notice the cancellation trend: the numerator of the first term ($5$) cancels out with the denominator of the second term ($5$). The numerator of the second term ($7$) cancels out with the denominator of the third term ($7$), and this continues through the entire chain. The surviving values are the first denominator ($3$) and the last numerator ($1001$): $$\frac{1001}{3}$$ ### Exam Strategy & Shortcut Observe the first two terms: $\frac{5}{3} \times \frac{7}{5} = \frac{7}{3}$. The result is always $\frac{\text{Last Numerator}}{\text{First Denominator}}$. Since the first denominator is $3$, look for options containing $3$ in the denominator or as a simplified fraction. Only option (d) matches this condition. ### Common Pitfall Students often invert the final ratio, arriving incorrectly at $\frac{3}{1001}$ due to losing track of which part of the fraction cancels out at the start and end of the series. ### Final Answer **Therefore, the correct answer is 1001/3.**