The simplified value of `(1-1/3) (1- 1/4) (1-1/5) …(1-1/(99)) (1- (1)/(100))` is

Options

A. `2/(99)`
B. `1/(25)`
C. `1/(50)`
D. `1/(100)`

Answer

To simplify the expression \((1 - \frac{1}{3})(1 - \frac{1}{4})(1 - \frac{1}{5}) \ldots (1 - \frac{1}{99})(1 - \frac{1}{100})\), we will follow these steps:

### Step 1: Rewrite each term
We start by rewriting each term in the product. The general term can be expressed as:
\[ 1 - \frac{1}{n} = \frac{n-1}{n} \]
Thus, we can rewrite the entire product:
\[ (1 - \frac{1}{3})(1 - \frac{1}{4})(1 - \frac{1}{5}) \ldots (1 - \frac{1}{99})(1 - \frac{1}{100}) = \frac{2}{3} \cdot \frac{3}{4} \cdot \frac{4}{5} \cdots \frac{98}{99} \cdot \frac{99}{100} \]

### Step 2: Formulate the product
Now, we can express the product as:
\[ \frac{2}{3} \cdot \frac{3}{4} \cdot \frac{4}{5} \cdots \frac{98}{99} \cdot \frac{99}{100} \]
Notice that in this product, most terms will cancel out.

### Step 3: Cancel terms
When we look closely, we can see that:
- The \(3\) in the numerator of the first fraction cancels with the \(3\) in the denominator of the second fraction.
- The \(4\) in the numerator of the second fraction cancels with the \(4\) in the denominator of the third fraction, and so on.

This cancellation continues all the way through to:
\[ \frac{2}{100} \]

### Step 4: Simplify the remaining fraction
Now we simplify \(\frac{2}{100}\):
\[ \frac{2}{100} = \frac{1}{50} \]

### Final Answer
Thus, the simplified value of the original expression is:
\[ \frac{1}{50} \]