Evaluate the telescoping product by writing each factor as a ratio: (1 + 1/2) * (1 + 1/3) * (1 + 1/4) * ... * (1 + 1/150) = ?

Introduction / Context:This question tests the technique of telescoping products in number theory and arithmetic simplification. By expressing each factor (1 + 1/n) as a simple fraction, most intermediate terms cancel, leaving only the first denominator and the last numerator. Recognizing this pattern saves time and avoids cumbersome multiplication.

Given Data / Assumptions:

• Product is (1 + 1/2)(1 + 1/3)(1 + 1/4)...(1 + 1/150).
• All factors are positive rational numbers.
• We can rearrange factors and simplify exactly (no rounding).

Concept / Approach:Write 1 + 1/n as (n + 1)/n. Then the entire product becomes a chain of fractions whose adjacent terms cancel (telescope). Only the first denominator and the last numerator remain after cancellation.

Step-by-Step Solution:

Verification / Alternative check:Compute a smaller sample, e.g., (1 + 1/2)(1 + 1/3)(1 + 1/4) = (3/2)(4/3)(5/4) = 5/2; the same cancellation pattern holds, confirming the approach.

Why Other Options Are Wrong:50.5, 65.5, 105, and 151 arise from partial telescoping or forgetting to divide by 2 at the end. Only 75.5 equals 151/2.

Common Pitfalls:Not converting to (n + 1)/n, multiplying straight across, or truncating the sequence boundaries incorrectly (starting at n = 1 would break the pattern).

Final Answer:75.5