Telescoping Product of the Form (1 − 1/k) Compute the product (1 − 1/3) (1 − 1/4) (1 − 1/5) … (1 − 1/n) for integer n ≥ 3.

Introduction / Context:Products of the form (1 − 1/k) are best handled by rewriting as simple fractions that telescope. This saves time and avoids expanding long products manually.

Given Data / Assumptions:

• Product index runs from k = 3 to k = n, where n ≥ 3.
• Each term simplifies to (k − 1)/k.

Concept / Approach:Rewrite each factor and cancel successive numerator–denominator pairs. Only the earliest numerator and the last denominator survive the cancellations, revealing a compact closed-form expression.

Step-by-Step Solution:(1 − 1/k) = (k − 1)/k.Product from k = 3 to n: Π (k − 1)/k = (2/3) * (3/4) * (4/5) * … * ((n − 1)/n).Telescoping leaves 2/n (all intermediate factors cancel).

Verification / Alternative check:Test n = 5: (1 − 1/3)(1 − 1/4)(1 − 1/5) = (2/3)(3/4)(4/5) = 2/5 = 2/n, confirming the formula.

Why Other Options Are Wrong:1/n misses the factor 2; 2(n − 1)/n and 2(n + 1)/n do not arise from telescoping; 1/(n − 1) is a common mis-cancellation error.

Common Pitfalls:Dropping the initial 2/3 factor; misaligning the cancellation ends; attempting to sum instead of multiplying.

Final Answer:2/n