If `x_(1)=2` and `x_(n+1)=sqrt(x_(n)^(2)+8)`, then `x_(4)=`

Options

A. `3.46`
B. `4.47`
C. `5.29`
D. `8.49`

Explanation

To solve for \( x_4 \) given the recursive formula \( x_{n+1} = \sqrt{x_n^2 + 8} \) and the initial value \( x_1 = 2 \), we will calculate each term step by step.

### Step 1: Calculate \( x_2 \)
Using the formula:
\[ x_2 = \sqrt{x_1^2 + 8} \]
Substituting \( x_1 = 2 \):
\[ x_2 = \sqrt{2^2 + 8} = \sqrt{4 + 8} = \sqrt{12} \]

### Step 2: Calculate \( x_3 \)
Now we use \( x_2 \) to find \( x_3 \):
\[ x_3 = \sqrt{x_2^2 + 8} \]
Substituting \( x_2 = \sqrt{12} \):
\[ x_3 = \sqrt{(\sqrt{12})^2 + 8} = \sqrt{12 + 8} = \sqrt{20} \]

### Step 3: Calculate \( x_4 \)
Now we use \( x_3 \) to find \( x_4 \):
\[ x_4 = \sqrt{x_3^2 + 8} \]
Substituting \( x_3 = \sqrt{20} \):
\[ x_4 = \sqrt{(\sqrt{20})^2 + 8} = \sqrt{20 + 8} = \sqrt{28} \]

### Final Result
Thus, the value of \( x_4 \) is:
\[ x_4 = \sqrt{28} \]
This can be simplified further:
\[ x_4 = \sqrt{4 \times 7} = 2\sqrt{7} \]
Numerically, \( \sqrt{28} \approx 5.29 \).

### Summary
The value of \( x_4 \) is \( \sqrt{28} \) or approximately \( 5.29 \).

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