If `x_(0) = 3 ` and `x_(n+1) = x_(n) sqrt(x_n + 1)` , then `x_(3) = `

Options

A. `15.9`
B. `31.7`
C. `44.9`
D. `65.2`

Explanation

To solve the problem, we need to compute the values of the sequence defined by the recurrence relation \( x_{n+1} = x_n \sqrt{x_n + 1} \) starting from \( x_0 = 3 \). We will calculate \( x_1 \), \( x_2 \), and \( x_3 \) step by step.

### Step 1: Calculate \( x_1 \)
Given:
- \( x_0 = 3 \)

Using the recurrence relation:
\[ x_1 = x_0 \sqrt{x_0 + 1} \]
Substituting the value of \( x_0 \):
\[ x_1 = 3 \sqrt{3 + 1} = 3 \sqrt{4} = 3 \times 2 = 6 \]

### Step 2: Calculate \( x_2 \)
Now, we need to calculate \( x_2 \):
\[ x_2 = x_1 \sqrt{x_1 + 1} \]
Substituting the value of \( x_1 \):
\[ x_2 = 6 \sqrt{6 + 1} = 6 \sqrt{7} \]

### Step 3: Calculate \( x_3 \)
Next, we calculate \( x_3 \):
\[ x_3 = x_2 \sqrt{x_2 + 1} \]
Substituting the value of \( x_2 \):
\[ x_3 = (6 \sqrt{7}) \sqrt{(6 \sqrt{7}) + 1} \]
First, we need to simplify \( 6 \sqrt{7} + 1 \):
\[ 6 \sqrt{7} + 1 \]
Now, we can calculate \( x_3 \):
\[ x_3 = 6 \sqrt{7} \sqrt{6 \sqrt{7} + 1} \]

### Step 4: Approximate \( x_3 \)
To find a numerical approximation, we can calculate:
1. \( \sqrt{7} \approx 2.64575 \)
2. \( 6 \sqrt{7} \approx 15.8745 \)
3. \( 6 \sqrt{7} + 1 \approx 15.8745 + 1 = 16.8745 \)
4. \( \sqrt{16.8745} \approx 4.109 \)
5. Finally, substituting back:
\[ x_3 \approx 6 \sqrt{7} \times 4.109 \approx 15.8745 \times 4.109 \approx 65.2 \]

Thus, the value of \( x_3 \) is approximately \( 65.2 \).

### Final Answer
\[ x_3 \approx 65.2 \]