(1-1/3)(1-1/4)(1-1/5)…(1-1/n) equals:

Options

A. `1/n`
B. `2/n`
C. `3/n`
D. `4/n`

Answer

To solve the expression \((1 - \frac{1}{3})(1 - \frac{1}{4})(1 - \frac{1}{5}) \ldots (1 - \frac{1}{n})\), 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 follows:
\[ 1 - \frac{1}{k} = \frac{k - 1}{k} \]
Therefore, we can express the entire product as:
\[ (1 - \frac{1}{3})(1 - \frac{1}{4})(1 - \frac{1}{5}) \ldots (1 - \frac{1}{n}) = \frac{2}{3} \cdot \frac{3}{4} \cdot \frac{4}{5} \cdots \frac{n-1}{n} \]

2. **Write the Full Expression:**
The product can be written as:
\[ \frac{2}{3} \cdot \frac{3}{4} \cdot \frac{4}{5} \cdots \frac{n-1}{n} \]

3. **Observe the Cancellation:**
Notice that in this product, all the intermediate terms will cancel out:
- 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 until the \(n-1\) in the numerator of \(\frac{n-1}{n}\) cancels with the \(n-1\) in the previous term.

4. **Final Simplification:**
After all cancellations, we are left with:
\[ \frac{2}{n} \]

Thus, the final result of the expression \((1 - \frac{1}{3})(1 - \frac{1}{4})(1 - \frac{1}{5}) \ldots (1 - \frac{1}{n})\) is:
\[ \frac{2}{n} \]