Determine the value of '?' in: `( 1- (1)/(2) ) ( 1- (1)/(3) ) ( 1- (1)/(4) ) ......... (1- ( 1)/( n) ) = ?`

Options

A. `2/n`
B. `1/n^2`
C. `1/n`
D. `2/n^2`

Answer

To solve the problem, we need to evaluate the expression:

\[ (1 - \frac{1}{2})(1 - \frac{1}{3})(1 - \frac{1}{4}) \ldots (1 - \frac{1}{n}) \]

### Step 1: Rewrite each term
We can rewrite each term in the product:

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

Thus, we can express the entire product as:

\[ (1 - \frac{1}{2})(1 - \frac{1}{3})(1 - \frac{1}{4}) \ldots (1 - \frac{1}{n}) = \frac{1}{2} \cdot \frac{2}{3} \cdot \frac{3}{4} \cdots \frac{n-1}{n} \]

### Step 2: Write the product in fraction form
Now, we can write this product as:

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

### Step 3: Simplify the expression
Notice that the numerator is \((n-1)!\) and the denominator is \(n!\):

\[ \frac{(n-1)!}{n!} = \frac{(n-1)!}{n \cdot (n-1)!} = \frac{1}{n} \]

### Conclusion
Thus, we find that:

\[ (1 - \frac{1}{2})(1 - \frac{1}{3})(1 - \frac{1}{4}) \ldots (1 - \frac{1}{n}) = \frac{1}{n} \]

The value of '?' is:

\[ \boxed{\frac{1}{n}} \]