The simplification of `(1-(1)/(2)) ( 1- (1)/(3)) ( 1-(1)/(4))...(1-(1)/(100))` results in :

Options

A. `0.01`
B. `0.001`
C. `1`
D. 0.1

Answer

To simplify the expression \((1 - \frac{1}{2})(1 - \frac{1}{3})(1 - \frac{1}{4}) \ldots (1 - \frac{1}{100})\), we can follow these steps:

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

### Step 2: Write Out the Product
Now, let's write out the product:
\[ \frac{1}{2} \cdot \frac{2}{3} \cdot \frac{3}{4} \cdots \frac{99}{100} \]
This can be expressed as:
\[ \frac{1 \cdot 2 \cdot 3 \cdots 99}{2 \cdot 3 \cdots 100} \]

### Step 3: Simplify the Product
Notice that in the numerator, we have \(1 \cdot 2 \cdot 3 \cdots 99\) which is \(99!\), and in the denominator, we have \(2 \cdot 3 \cdots 100\) which can be rewritten as:
\[ 2 \cdot 3 \cdots 100 = \frac{100!}{1} \]
Thus, we can simplify the expression to:
\[ \frac{99!}{100!} = \frac{99!}{100 \cdot 99!} = \frac{1}{100} \]

### Step 4: Final Result
Therefore, the final result of the simplification is:
\[ \frac{1}{100} = 0.01 \]