Three consecutive terms, in order, of an arithmetic sequence are `x+sqrt(2),2x+sqrt(3), and 5x-sqrt(5)`. Then x equals

Options

A. 2.14
B. 2.45
C. 2.46
D. 3.24

Explanation

To solve the problem, we need to find the value of \( x \) given the three consecutive terms of an arithmetic sequence: \( x + \sqrt{2} \), \( 2x + \sqrt{3} \), and \( 5x - \sqrt{5} \).

### Step-by-Step Solution:

1. **Identify the Terms**:
The three terms are:
- First term: \( a_1 = x + \sqrt{2} \)
- Second term: \( a_2 = 2x + \sqrt{3} \)
- Third term: \( a_3 = 5x - \sqrt{5} \)

2. **Use the Property of Arithmetic Sequence**:
In an arithmetic sequence, the difference between consecutive terms is constant. Therefore, we can set up the equation:
\[ a_2 - a_1 = a_3 - a_2 \]

3. **Calculate \( a_2 - a_1 \)**:
\[ a_2 - a_1 = (2x + \sqrt{3}) - (x + \sqrt{2}) = 2x + \sqrt{3} - x - \sqrt{2} = x + \sqrt{3} - \sqrt{2} \]

4. **Calculate \( a_3 - a_2 \)**:
\[ a_3 - a_2 = (5x - \sqrt{5}) - (2x + \sqrt{3}) = 5x - \sqrt{5} - 2x - \sqrt{3} = 3x - \sqrt{5} - \sqrt{3} \]

5. **Set the Two Differences Equal**:
Now we equate the two differences:
\[ x + \sqrt{3} - \sqrt{2} = 3x - \sqrt{5} - \sqrt{3} \]

6. **Rearrange the Equation**:
Move all terms involving \( x \) to one side and constant terms to the other side:
\[ x - 3x = -\sqrt{5} - \sqrt{3} - \sqrt{3} + \sqrt{2} \]
Simplifying gives:
\[ -2x = -\sqrt{5} - 2\sqrt{3} + \sqrt{2} \]

7. **Solve for \( x \)**:
Divide both sides by -2:
\[ x = \frac{\sqrt{5} + 2\sqrt{3} - \sqrt{2}}{2} \]

### Final Answer:
Thus, the value of \( x \) is:
\[ x = \frac{\sqrt{5} + 2\sqrt{3} - \sqrt{2}}{2} \]