Telescoping Product with Odd Fractions Evaluate the product (2 − 1/3) (2 − 3/5) (2 − 5/7) … (2 − 997/999).

Introduction / Context:
This product has a telescoping structure. Each factor is of the form 2 − n/(n + 2) with n running over odd numbers 1, 3, 5, …, 997. Recognizing and converting each factor to a simple ratio produces massive cancellation.

Given Data / Assumptions:

• General term: 2 − n/(n + 2), where n takes odd values from 1 to 997.
• Product spans consecutive odd n, so adjacent terms share canceling numerators and denominators after simplification.

Concept / Approach:
Rewrite each factor: 2 − n/(n + 2) = (2(n + 2) − n)/(n + 2) = (n + 4)/(n + 2). The product becomes Π (n + 4)/(n + 2). Because n increases by 2 each step, most factors cancel.

Step-by-Step Solution:
Transform factors: (2 − n/(n + 2)) → (n + 4)/(n + 2).Sequence of numerators: 5, 7, 9, …, 1001.Sequence of denominators: 3, 5, 7, …, 999.Telescoping cancellation leaves only 1001/3.

Verification / Alternative check:
Check initial and final fringes: first denominator 3 remains in the bottom; last numerator 1001 remains on top; every intermediate 5, 7, …, 999 cancels.

Why Other Options Are Wrong:
5/999 and 1001/999 keep extraneous terms; “None of these” is incorrect since a clean closed form 1001/3 exists; 1000/3 is off by 1 from the final numerator.

Common Pitfalls:
Failing to recognize the (n + 4)/(n + 2) representation; truncating the product incorrectly at endpoints; arithmetic slips with large boundary values.

Final Answer:
1001/3