`[(1-1/3)(1-1/4)(1-1/5)(1-1/6)....(1-1/99)(1-1/100)]=?`

Options

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

Answer

To solve the expression \([(1 - \frac{1}{3})(1 - \frac{1}{4})(1 - \frac{1}{5})(1 - \frac{1}{6}) \ldots (1 - \frac{1}{99})(1 - \frac{1}{100})]\), we can simplify each term step by step.

### Step-by-Step Solution:

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

2. **Form the Product**:
The product can be expressed as:
\[ \frac{2}{3} \cdot \frac{3}{4} \cdot \frac{4}{5} \cdots \frac{99}{100} \]
Notice that in this product, all the intermediate terms will cancel out.

3. **Canceling Terms**:
When we multiply these fractions, we can see that:
- 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:
- The \(99\) in the numerator of \(\frac{98}{99}\) cancels with the \(99\) in the denominator of \(\frac{99}{100}\).

Therefore, we are left with:
\[ \frac{2}{100} \]

4. **Simplify the Result**:
Now, simplify \(\frac{2}{100}\):
\[ \frac{2}{100} = \frac{1}{50} \]

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