$\left[ 2 - \frac{1}{3} ight] \left[ 2 - \frac{3}{5} ight] \left[ 2 - \frac{5}{7} ight] \dots \dots \dots \left[ 2 - \frac{997}{999} ight] ext{ is :}$

Question

$\left[ 2 - \frac{1}{3} ight] \left[ 2 - \frac{3}{5} ight] \left[ 2 - \frac{5}{7} ight] \dots \dots \dots \left[ 2 - \frac{997}{999} ight] 	ext{ is :}$

Accepted Answer

To solve the given problem, we need to evaluate the product:

$$
\left( 2 - \frac{1}{3} \right) \left( 2 - \frac{3}{5} \right) \left( 2 - \frac{5}{7} \right) \dots \left( 2 - \frac{997}{999} \right)
$$

Each term in the product can be rewritten as:

$$
2 - \frac{n}{n+2} = \frac{2(n+2) - n}{n+2} = \frac{2n + 4 - n}{n+2} = \frac{n + 4}{n+2}
$$

Thus, the product becomes:

$$
\prod_{n=1, 3, 5, \ldots, 997} \frac{n+4}{n+2}
$$

This is a telescoping product. Let's write out the first few terms to see the pattern:

- For $ n = 1 $, the term is $ \frac{5}{3} $
- For $ n = 3 $, the term is $ \frac{7}{5} $
- For $ n = 5 $, the term is $ \frac{9}{7} $
- ...
- For $ n = 997 $, the term is $ \frac{1001}{999} $

Notice that in this telescoping product, most terms will cancel out. Specifically, the denominator of each fraction cancels with the numerator of the next fraction. Therefore, the product simplifies to:

$$
\frac{5}{3} \times \frac{1001}{999}
$$

The remaining terms are the first numerator and the last denominator. Therefore, the entire product simplifies to:

$$
\frac{1001}{3}
$$

Thus, the correct answer is Option 3: $\frac{1001}{3}$.

**Explanation of Incorrect Options:**

1. **Option 1: $\frac{5}{999}$** - This option incorrectly assumes that the product simplifies to the first numerator over the last denominator, without considering the telescoping nature of the product.

2. **Option 2: $\frac{1001}{999}$** - This option represents only the last term of the product, ignoring the initial term $\frac{5}{3}$ that remains after cancellation.

3. **Option 4: $\frac{1001}{333}$** - This option does not correctly account for the telescoping cancellation and the resulting simplification of the product.

Therefore, the correct answer is indeed Option 3: $\frac{1001}{3}$.