Question

Find the value of \( (1-\frac{1}{3})(1-\frac{1}{4})(1-\frac{1}{5})\ldots(1-\frac{1}{100}) \)

• A. \(\frac{1}{25}\)
• B. \(\frac{1}{50}\)
• C. \(\frac{1}{15}\)
• D. \(\frac{1}{45}\)
• E.

Hint:

For products of sequences, identify any patterns or cancellation opportunities to simplify the calculations.

Answer 1

The Correct Option is B

Step 1: The given expression is: \[ \left( 1 - \frac{1}{3} \right) \left( 1 - \frac{1}{4} \right) \left( 1 - \frac{1}{5} \right) \ldots \left( 1 - \frac{1}{100} \right) \] \[ = \frac{2}{3} \times \frac{3}{4} \times \frac{4}{5} \times \ldots \times \frac{99}{100} \] Step 2: Notice that most terms in the product cancel out, resulting in: \[ \frac{2}{100} \] Step 3: Simplifying this expression gives: \[ \frac{2}{100} = \frac{1}{50} \] Thus, the correct answer is \( \frac{1}{50} \).