If $$ (4-x) ; 3 $$ and $$ \sqrt{2 x+5} $$ are the first three ter if an anthmetic Serquence a). Determine the value of $$ x $$.

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To determine the value of $$ x $$ such that $$ (4 - x) $$, $$ 3 $$, and $$ \sqrt{2x + 5} $$ are the first three terms of an arithmetic sequence, follow these steps:

### Step 1: Understand the Arithmetic Sequence Condition
In an arithmetic sequence, the difference between consecutive terms is constant. Therefore:

$$
\text{Second term} - \text{First term} = \text{Third term} - \text{Second term}
$$

Substituting the given terms:

$$
3 - (4 - x) = \sqrt{2x + 5} - 3
$$

### Step 2: Simplify the Equation
$$
3 - 4 + x = \sqrt{2x + 5} - 3
$$
$$
-1 + x = \sqrt{2x + 5} - 3
$$

Add 3 to both sides to isolate the square root:

$$
x + 2 = \sqrt{2x + 5}
$$

### Step 3: Solve for $$ x $$
Square both sides to eliminate the square root:

$$
(x + 2)^2 = 2x + 5
$$
$$
x^2 + 4x + 4 = 2x + 5
$$
$$
x^2 + 2x - 1 = 0
$$

Solve the quadratic equation using the quadratic formula:

$$
x = \frac{-2 \pm \sqrt{(2)^2 - 4(1)(-1)}}{2(1)} = \frac{-2 \pm \sqrt{8}}{2} = \frac{-2 \pm 2\sqrt{2}}{2} = -1 \pm \sqrt{2}
$$

### Step 4: Verify Potential Solutions
1. **For $$ x = -1 + \sqrt{2} $$:**
   - Compute $$ \sqrt{2x + 5} = \sqrt{2(-1 + \sqrt{2}) + 5} = \sqrt{3 + 2\sqrt{2}} = 1 + \sqrt{2} $$
   - The sequence becomes $$ 5 - \sqrt{2} $$, $$ 3 $$, $$ 1 + \sqrt{2} $$
   - The differences: $$ 3 - (5 - \sqrt{2}) = -2 + \sqrt{2} $$ and $$ (1 + \sqrt{2}) - 3 = -2 + \sqrt{2} $$
   - The differences are equal, so $$ x = -1 + \sqrt{2} $$ is valid.

2. **For $$ x = -1 - \sqrt{2} $$:**
   - Compute $$ \sqrt{2x + 5} = \sqrt{3 - 2\sqrt{2}} = \sqrt{2} - 1 $$
   - The sequence becomes $$ 5 + \sqrt{2} $$, $$ 3 $$, $$ \sqrt{2} - 1 $$
   - The differences: $$ 3 - (5 + \sqrt{2}) = -2 - \sqrt{2} $$ and $$ (\sqrt{2} - 1) - 3 = \sqrt{2} - 4 $$
   - The differences are not equal, so $$ x = -1 - \sqrt{2} $$ is extraneous.

### Conclusion
The valid solution is:

$$
x = \sqrt{2} - 1
$$

**Answer:** After simplifying, the valid solution is x = √2 minus one. In symbols,

x = √2 − 1
The value of $$ x $$ is $$ \sqrt{2} - 1 $$.

If \( (4-x) ; 3 \) and \( \sqrt{2 x+5} \) are the first three ter if an anthmetic Serquence a). Determine the value of \( x \).

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