Telescoping product over odd terms Compute (2 − 1/3)(2 − 3/5)(2 − 5/7)…(2 − 999/1001) exactly.

Introduction / Context:
Telescoping products are powerful when each factor can be expressed in a form that cancels with neighboring terms. Recognizing the general term and its simplification unlocks a dramatic reduction in effort and avoids massive multiplications.

Given Data / Assumptions:

• Factors: (2 − 1/3), (2 − 3/5), (2 − 5/7), …, (2 − 999/1001).
• The pattern uses odd numerators k and denominators k + 2, with k = 1, 3, 5, …, 999.

Concept / Approach:
For a general odd k, 2 − k/(k + 2) = (2(k + 2) − k)/(k + 2) = (2k + 4 − k)/(k + 2) = (k + 4)/(k + 2). The product over successive odd k thus becomes a chain of fractions whose numerators cancel with later denominators, forming a telescoping product.

Step-by-Step Solution:
Write each factor as (k + 4)/(k + 2).Sequence samples: for k = 1: 5/3; k = 3: 7/5; k = 5: 9/7; … ; k = 999: 1003/1001.Multiplying cancels intermediate 5, 7, 9, …, 1001 between numerators and denominators.Product reduces to final numerator over initial denominator: 1003/3.

Verification / Alternative check:
Compute the first few and last few factors to observe the cancellation pattern explicitly. The only surviving terms are the very first denominator (3) and the very last numerator (1003).

Why Other Options Are Wrong:
991/1001 and 1001/13 do not follow the correct telescoping endpoints.“None of these” is incorrect because 1003/3 is obtained exactly.

Common Pitfalls:
Misidentifying the general factor; starting k at 0 or using even k; or forgetting that only the first denominator and last numerator remain after cancellation.

Final Answer:
1003/3