When simplified the product `(1-1/3)(1-1/4)(1-1/5)......(1-1/n)`
becomes :

Options

A. `1/n`
B. `2/n`
C. `(2(n-1))/n`
D. `2/(n(n+1))`

Answer

To simplify the product \((1 - \frac{1}{3})(1 - \frac{1}{4})(1 - \frac{1}{5}) \ldots (1 - \frac{1}{n})\), we can follow these steps:

### Step 1: Rewrite each term in the product
Each term in the product can be rewritten as follows:

\[ 1 - \frac{1}{k} = \frac{k - 1}{k} \]

where \(k\) takes values from 3 to \(n\). Therefore, we can express the entire product as:

\[ \prod_{k=3}^{n} \left(1 - \frac{1}{k}\right) = \prod_{k=3}^{n} \frac{k - 1}{k} \]

### Step 2: Expand the product
Now, we can expand the product:

\[ \prod_{k=3}^{n} \frac{k - 1}{k} = \frac{2}{3} \cdot \frac{3 - 1}{4} \cdot \frac{4 - 1}{5} \cdots \frac{n - 1}{n} \]

This can be simplified as:

\[ = \frac{2}{3} \cdot \frac{2}{4} \cdot \frac{3}{5} \cdots \frac{n - 1}{n} \]

### Step 3: Identify the pattern
Notice that in the product, the numerator of each fraction cancels with the denominator of the next fraction. Thus, we can see that:

\[ = \frac{2}{n} \]

### Step 4: Final result
Therefore, the simplified product is:

\[ \frac{2}{n} \]

### Conclusion
The final result of the product \((1 - \frac{1}{3})(1 - \frac{1}{4})(1 - \frac{1}{5}) \ldots (1 - \frac{1}{n})\) simplifies to \(\frac{2}{n}\).