simplified value of `(1-1/3)(1-1/4)(1-1/5)......(1-1/99 )(1-1/100)` is

Options

A. `2/99`
B. `1/25`
C. `1/50`
D. `1/100`

Answer

To simplify the expression \((1 - \frac{1}{3})(1 - \frac{1}{4})(1 - \frac{1}{5}) \ldots (1 - \frac{1}{99})(1 - \frac{1}{100})\), we will follow these steps:

1. **Rewrite Each Term**:
Each term in the product can be rewritten as:
\[ 1 - \frac{1}{n} = \frac{n - 1}{n} \]
Therefore, we can express the entire product as:
\[ (1 - \frac{1}{3})(1 - \frac{1}{4})(1 - \frac{1}{5}) \ldots (1 - \frac{1}{99})(1 - \frac{1}{100}) = \frac{2}{3} \cdot \frac{3}{4} \cdot \frac{4}{5} \cdots \frac{98}{99} \cdot \frac{99}{100} \]

2. **Write the Full Expression**:
The full expression can be written as:
\[ \frac{2}{3} \cdot \frac{3}{4} \cdot \frac{4}{5} \cdots \frac{98}{99} \cdot \frac{99}{100} \]

3. **Cancel Terms**:
In this expression, we notice that many terms will cancel out. Specifically, every numerator will cancel with the denominator of the next term:
- The \(3\) in the numerator of \(\frac{2}{3}\) cancels with the \(3\) in the denominator of \(\frac{3}{4}\).
- The \(4\) in the numerator of \(\frac{3}{4}\) cancels with the \(4\) in the denominator of \(\frac{4}{5}\).
- This pattern continues all the way up to \(\frac{99}{100}\).

4. **Identify Remaining Terms**:
After canceling, we are left with:
\[ \frac{2}{100} \]

5. **Simplify the Remaining Fraction**:
Now we simplify \(\frac{2}{100}\):
\[ \frac{2}{100} = \frac{1}{50} \]

Thus, the simplified value of the expression is:
\[ \frac{1}{50} \]