The sequence `{x_(k)}` is defined by `x_(k+1)=x_(k)^(2)+x_(k)` and `x_(1)=(1)/(2)`. Then `[(1)/(x_(1)+1)+(1)/(x_(2)+1)+...+(1)/(x_(100)+1)]` (where `[.]` denotes the greatest integer function) is equal to

Options

A. `0`
B. `2`
C. `4`
D. `1`

Explanation

To solve the problem step by step, we will analyze the sequence defined by \( x_{k+1} = x_k^2 + x_k \) with the initial condition \( x_1 = \frac{1}{2} \). We need to compute the sum \( \left\lfloor \frac{1}{x_1 + 1} + \frac{1}{x_2 + 1} + \cdots + \frac{1}{x_{100} + 1} \right\rfloor \).

### Step 1: Calculate the first few terms of the sequence

1. **Calculate \( x_1 \)**:
\[ x_1 = \frac{1}{2} \]

2. **Calculate \( x_2 \)**:
\[ x_2 = x_1^2 + x_1 = \left(\frac{1}{2}\right)^2 + \frac{1}{2} = \frac{1}{4} + \frac{1}{2} = \frac{1}{4} + \frac{2}{4} = \frac{3}{4} \]

3. **Calculate \( x_3 \)**:
\[ x_3 = x_2^2 + x_2 = \left(\frac{3}{4}\right)^2 + \frac{3}{4} = \frac{9}{16} + \frac{12}{16} = \frac{21}{16} \]

4. **Calculate \( x_4 \)**:
\[ x_4 = x_3^2 + x_3 = \left(\frac{21}{16}\right)^2 + \frac{21}{16} = \frac{441}{256} + \frac{336}{256} = \frac{777}{256} \]

5. **Calculate \( x_5 \)**:
\[ x_5 = x_4^2 + x_4 = \left(\frac{777}{256}\right)^2 + \frac{777}{256} = \frac{604929}{65536} + \frac{199488}{65536} = \frac{804417}{65536} \]

### Step 2: Calculate \( \frac{1}{x_k + 1} \) for the first few terms

1. **Calculate \( \frac{1}{x_1 + 1} \)**:
\[ \frac{1}{x_1 + 1} = \frac{1}{\frac{1}{2} + 1} = \frac{1}{\frac{3}{2}} = \frac{2}{3} \]

2. **Calculate \( \frac{1}{x_2 + 1} \)**:
\[ \frac{1}{x_2 + 1} = \frac{1}{\frac{3}{4} + 1} = \frac{1}{\frac{7}{4}} = \frac{4}{7} \]

3. **Calculate \( \frac{1}{x_3 + 1} \)**:
\[ \frac{1}{x_3 + 1} = \frac{1}{\frac{21}{16} + 1} = \frac{1}{\frac{37}{16}} = \frac{16}{37} \]

4. **Calculate \( \frac{1}{x_4 + 1} \)**:
\[ \frac{1}{x_4 + 1} = \frac{1}{\frac{777}{256} + 1} = \frac{1}{\frac{1033}{256}} = \frac{256}{1033} \]

5. **Calculate \( \frac{1}{x_5 + 1} \)**:
\[ \frac{1}{x_5 + 1} = \frac{1}{\frac{804417}{65536} + 1} = \frac{65536}{804417 + 65536} = \frac{65536}{869953} \]

### Step 3: Sum the first few terms

Now we can sum these values:
\[ S = \frac{2}{3} + \frac{4}{7} + \frac{16}{37} + \frac{256}{1033} + \frac{65536}{869953} + \cdots \]

### Step 4: Estimate the total sum

The values of \( \frac{1}{x_k + 1} \) decrease rapidly as \( k \) increases. The first few terms contribute significantly to the sum, while the later terms contribute very little.

Calculating the approximate values:
- \( \frac{2}{3} \approx 0.6667 \)
- \( \frac{4}{7} \approx 0.5714 \)
- \( \frac{16}{37} \approx 0.4324 \)
- \( \frac{256}{1033} \approx 0.2473 \)
- \( \frac{65536}{869953} \approx 0.0754 \)

Adding these:
\[ S \approx 0.6667 + 0.5714 + 0.4324 + 0.2473 + 0.0754 \approx 1.9932 \]

### Step 5: Apply the greatest integer function

Finally, we apply the greatest integer function:
\[ \left\lfloor S \right\rfloor = \left\lfloor 1.9932 \right\rfloor = 1 \]

### Final Answer
Thus, the answer is:
\[ \boxed{1} \]