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)/(1000`

Answer

To simplify the expression \( S = (1 - \frac{1}{3})(1 - \frac{1}{4})(1 - \frac{1}{5}) \cdots (1 - \frac{1}{100}) \), we can follow these steps:

### Step 1: Rewrite each term
Each term in the product can be rewritten:
\[ 1 - \frac{1}{n} = \frac{n - 1}{n} \]
Thus, we can express the entire product as:
\[ S = \frac{2}{3} \cdot \frac{3}{4} \cdot \frac{4}{5} \cdots \frac{99}{100} \]

### Step 2: Write the product in a clearer form
Now we can write the product more clearly:
\[ S = \frac{2}{3} \cdot \frac{3}{4} \cdot \frac{4}{5} \cdots \frac{99}{100} = \frac{2 \cdot 3 \cdot 4 \cdots 99}{3 \cdot 4 \cdots 100} \]

### Step 3: Cancel common terms
Notice that in the numerator and denominator, many terms will cancel out:
- The \(3\) in the numerator cancels with the \(3\) in the denominator.
- The \(4\) in the numerator cancels with the \(4\) in the denominator.
- This pattern continues up to \(99\).

After canceling, we are left with:
\[ S = \frac{2}{100} \]

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

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

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