If `f(x)=1-f(1-x)`, then the value of
`f((1)/(999))+f((2)/(999))+...+f((998)/(999))` is :

Options

A. a. 998
B. b. 1
C. c. 499
D. d. none of these

Answer

To solve the problem given \( f(x) = 1 - f(1 - x) \), we need to find the value of

\[ f\left(\frac{1}{999}\right) + f\left(\frac{2}{999}\right) + \ldots + f\left(\frac{998}{999}\right). \]

### Step-by-Step Solution:

1. **Understanding the Function**:
We have the functional equation:
\[ f(x) + f(1 - x) = 1. \]
This means that for any \( x \), \( f(x) \) and \( f(1 - x) \) always sum to 1.

2. **Pairing the Terms**:
We can pair the terms in the sum:
- The first term is \( f\left(\frac{1}{999}\right) \) and the last term is \( f\left(\frac{998}{999}\right) \).
- The second term is \( f\left(\frac{2}{999}\right) \) and the second last term is \( f\left(\frac{997}{999}\right) \).
- Continuing this way, we can pair:
\[ f\left(\frac{k}{999}\right) + f\left(\frac{999-k}{999}\right) = 1 \quad \text{for } k = 1, 2, \ldots, 498. \]

3. **Counting the Pairs**:
Each pair sums to 1. Since we have terms from \( k = 1 \) to \( k = 998 \), we can form pairs:
- There are \( 998 \) terms in total.
- The number of pairs is \( \frac{998}{2} = 499 \).

4. **Calculating the Total**:
Since each of the \( 499 \) pairs sums to \( 1 \), the total sum is:
\[ 499 \times 1 = 499. \]

### Final Answer:
Thus, the value of

\[ f\left(\frac{1}{999}\right) + f\left(\frac{2}{999}\right) + \ldots + f\left(\frac{998}{999}\right) = 499. \]