`(2-1/3)(2-3/5)(2-5/7).......(2-997/999)=?`

Options

A. `5/999`
B. `1001/999`
C. `1001/3`
D. None of these

Answer

To solve the expression \((2 - \frac{1}{3})(2 - \frac{3}{5})(2 - \frac{5}{7}) \ldots (2 - \frac{997}{999})\), we will simplify each term step by step.

### Step 1: Rewrite Each Term
Each term in the product can be rewritten as follows:
\[ 2 - \frac{n}{m} = \frac{2m - n}{m} \]
where \(n\) takes values from 1 to 997 and \(m\) takes values from 3 to 999 in odd increments.

### Step 2: Identify the Pattern
The terms can be expressed as:
- For \(n = 1\) and \(m = 3\): \(2 - \frac{1}{3} = \frac{6 - 1}{3} = \frac{5}{3}\)
- For \(n = 3\) and \(m = 5\): \(2 - \frac{3}{5} = \frac{10 - 3}{5} = \frac{7}{5}\)
- For \(n = 5\) and \(m = 7\): \(2 - \frac{5}{7} = \frac{14 - 5}{7} = \frac{9}{7}\)

Continuing this pattern, we can generalize:
\[ 2 - \frac{n}{m} = \frac{2m - n}{m} \]

### Step 3: Write the Full Product
The full product can be expressed as:
\[ \prod_{k=1}^{499} \left(2 - \frac{2k - 1}{2k + 1}\right) = \prod_{k=1}^{499} \frac{(2(2k + 1) - (2k - 1))}{(2k + 1)} = \prod_{k=1}^{499} \frac{(4k + 2 - 2k + 1)}{(2k + 1)} \]
This simplifies to:
\[ \prod_{k=1}^{499} \frac{(2k + 3)}{(2k + 1)} \]

### Step 4: Simplify the Product
Notice that this product can be simplified:
\[ \prod_{k=1}^{499} \frac{(2k + 3)}{(2k + 1)} = \frac{3}{1} \cdot \frac{5}{3} \cdot \frac{7}{5} \cdots \frac{999}{997} \]
Most terms will cancel out, leaving:
\[ \frac{1001}{3} \]

### Final Answer
Thus, the value of the entire expression is:
\[ \frac{1001}{3} \]