The value of `(1-1/2)(1-1/3)(1-1/4)(1-1/5)…(1-1/n) is :

Options

A. 1
B. `(1-1/n)^5`
C. `1/n`
D. can't be determined

Answer

To solve the expression \((1 - \frac{1}{2})(1 - \frac{1}{3})(1 - \frac{1}{4})(1 - \frac{1}{5}) \ldots (1 - \frac{1}{n})\), let's break it down step by step.

### Step 1: Simplify Each Term
We will simplify each term in the product:
- For \(1 - \frac{1}{2}\):
\[ 1 - \frac{1}{2} = \frac{1}{2} \]
- For \(1 - \frac{1}{3}\):
\[ 1 - \frac{1}{3} = \frac{2}{3} \]
- For \(1 - \frac{1}{4}\):
\[ 1 - \frac{1}{4} = \frac{3}{4} \]
- For \(1 - \frac{1}{5}\):
\[ 1 - \frac{1}{5} = \frac{4}{5} \]
- Continuing this pattern, for \(1 - \frac{1}{n}\):
\[ 1 - \frac{1}{n} = \frac{n-1}{n} \]

### Step 2: Write the Complete Product
Now, we can write the complete product:
\[ (1 - \frac{1}{2})(1 - \frac{1}{3})(1 - \frac{1}{4}) \ldots (1 - \frac{1}{n}) = \frac{1}{2} \cdot \frac{2}{3} \cdot \frac{3}{4} \cdot \frac{4}{5} \cdots \frac{n-1}{n} \]

### Step 3: Observe the Pattern
Notice that in the product, most terms will cancel out:
- The numerator of each fraction cancels with the denominator of the next fraction.

### Step 4: Write the Canceled Form
After cancellation, we are left with:
\[ \frac{1}{n} \]

### Final Answer
Thus, the value of the expression \((1 - \frac{1}{2})(1 - \frac{1}{3})(1 - \frac{1}{4}) \ldots (1 - \frac{1}{n})\) is:
\[ \frac{1}{n} \]