On simplification of the following, the result will be: (1 - 1/2)(1 - 1/3)(1 -…

Concept

For a product of consecutive terms of the form (1 - 1/n), each factor simplifies to (n-1)/n. Multiplying a chain of such fractions produces a telescoping cancellation: every numerator except the very first cancels with the denominator of the previous fraction, so only the first numerator and the last denominator survive.

Application

1. Rewrite each factor as a single fraction: (1 - 1/2) = 1/2, (1 - 1/3) = 2/3, (1 - 1/4) = 3/4, and so on up to (1 - 1/100) = 99/100.

2. Write out the full product: (1/2) × (2/3) × (3/4) × ... × (99/100).

3. Cancel telescopically: the 2 in the numerator of the second fraction cancels with the 2 in the denominator of the first fraction; the 3 in the numerator of the third fraction cancels with the 3 in the denominator of the second fraction; and this pattern continues all the way through 99.

4. After every interior term cancels, only the first numerator (1) and the last denominator (100) remain, giving 1/100.

5. Convert the surviving fraction to decimal form: 1/100 = 0.01.

Cross-check

Check the pattern on a shorter version of the same product: (1 - 1/2)(1 - 1/3) = (1/2)(2/3) = 1/3, which matches the rule of “first numerator over last denominator” (1 over 3). Applying the same rule all the way up to (1 - 1/100) confirms the result is 1 over 100, that is, 0.01.