Evaluate the telescoping product (1 − 1/2)(1 − 1/3)(1 − 1/4)…(1 − 1/m) for an integer m ≥ 2. Express the result in simplest terms.

Introduction / Context:Products constructed as (1 − 1/k) from k = 2 to m are engineered to telescope when written as fractions with consecutive numerators and denominators. Recognizing this pattern allows a one-line simplification.

Given Data / Assumptions:

• Product: ∏ from k = 2 to m of (1 − 1/k), where m ≥ 2.
• All factors are positive fractions between 0 and 1.

Concept / Approach:Rewrite each term as (k − 1)/k. The product then becomes (1/2) * (2/3) * (3/4) * … * ((m − 1)/m). Consecutive numerator–denominator pairs cancel across the chain, leaving only the first numerator and the last denominator.

Step-by-Step Solution:(1 − 1/2)(1 − 1/3)…(1 − 1/m) = (1/2)(2/3)…((m − 1)/m).Telescoping cancellation: all intermediate numbers 2, 3, …, m − 1 cancel.Remaining fraction = 1/m.

Verification / Alternative check:Try small m: for m = 4, product = (1/2)(2/3)(3/4) = 1/4 = 1/m, confirming the general result.

Why Other Options Are Wrong:

• m, m + 1: These ignore that each factor is less than 1; the product cannot exceed 1.
• 1/(m − 1), 1/(m + 1): Common slips from stopping the telescoping one term too early or late.

Common Pitfalls:Failing to convert to (k − 1)/k; arithmetic errors in partial products that obscure the telescoping structure.

Final Answer:1/m