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
Each term in the product can be rewritten. For example:
\[ 1 - \frac{1}{k} = \frac{k-1}{k} \]
Thus, we can rewrite the entire product as:
\[ (1 - \frac{1}{3})(1 - \frac{1}{4})(1 - \frac{1}{5}) \ldots (1 - \frac{1}{n}) = \frac{2}{3} \cdot \frac{3}{4} \cdot \frac{4}{5} \cdots \frac{n-1}{n} \]

### Step 2: Express the Product
Now, we can express the product in a more compact form:
\[ \frac{2}{3} \cdot \frac{3}{4} \cdot \frac{4}{5} \cdots \frac{n-1}{n} = \frac{2}{n} \]
This is because all intermediate terms will cancel out.

### Step 3: Final Expression
Thus, the simplified form of the product is:
\[ \frac{2}{n} \]

### Solution Summary
The final answer is:
\[ \frac{2}{n} \]

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