$$ 3-t ;-t ; \sqrt{9-2 t} $$ are the first three terms of an arithmetic sequence. 3.1 Determine the value of $$ t $$.

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To determine the value of $$ t $$ such that $$ 3 - t $$, $$ -t $$, and $$ \sqrt{9 - 2t} $$ form an arithmetic sequence, follow these steps:

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

$$
(-t) - (3 - t) = \sqrt{9 - 2t} - (-t)
$$

### 2. **Simplify the Equation:**

$$
-t - 3 + t = \sqrt{9 - 2t} + t
$$

$$
-3 = \sqrt{9 - 2t} + t
$$

### 3. **Solve for $$ \sqrt{9 - 2t} $$:**

$$
\sqrt{9 - 2t} = -3 - t
$$

**Important Considerations:**
- The expression inside the square root must be non-negative:
  $$
  9 - 2t \geq 0 \implies t \leq 4.5
  $$
- The right side of the equation $$ \sqrt{9 - 2t} = -3 - t $$ must be non-negative:
  $$
  -3 - t \geq 0 \implies t \leq -3
  $$

### 4. **Square Both Sides to Eliminate the Square Root:**

$$
( \sqrt{9 - 2t} )^2 = (-3 - t)^2
$$

$$
9 - 2t = 9 + 6t + t^2
$$

$$
0 = t^2 + 8t
$$

$$
t(t + 8) = 0
$$

**Possible Solutions:**
- $$ t = 0 $$
- $$ t = -8 $$

### 5. **Verify the Solutions:**
- **For $$ t = 0 $$:**
  - Sequence: $$ 3 $$, $$ 0 $$, $$ 3 $$
  - Differences: $$ 0 - 3 = -3 $$ and $$ 3 - 0 = 3 $$ (Not constant)
  - **Rejected**

- **For $$ t = -8 $$:**
  - Sequence: $$ 11 $$, $$ 8 $$, $$ 5 $$
  - Differences: $$ 8 - 11 = -3 $$ and $$ 5 - 8 = -3 $$ (Constant)
  - **Accepted**

### **Conclusion:**
The value of $$ t $$ that satisfies the condition is:

$$
\boxed{-8}
$$
$$ t = -8 $$

are the first three terms of an arithmetic sequence.
3.1 Determine the value of .

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are the first three terms of an arithmetic sequence. 3.1 Determine the value of .