3. The first three terms of the sequence $$ 6 ; x ; y ; 27 $$ form an arithmetic progression and the last three terms form a geometric progression. 3.1 Determine the values of $$ x $$ and $$ y $$. 4.2 State the arithmetic and geometric sequences for each of your solutions.

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### Problem 3.1: Determine the values of $$ x $$ and $$ y $$.

Given the sequence $$ 6 ; x ; y ; 27 $$:

1. **Arithmetic Progression (AP) for the first three terms**:
   
   For an arithmetic progression, the difference between consecutive terms is constant. Therefore:
   $$
   x - 6 = y - x \implies 2x = 6 + y \implies y = 2x - 6
   $$

2. **Geometric Progression (GP) for the last three terms**:
   
   For a geometric progression, the ratio between consecutive terms is constant. Therefore:
   $$
   \frac{y}{x} = \frac{27}{y} \implies y^2 = 27x
   $$

3. **Substituting $$ y = 2x - 6 $$ into the GP equation**:
   $$
   (2x - 6)^2 = 27x
   $$
   $$
   4x^2 - 24x + 36 = 27x
   $$
   $$
   4x^2 - 51x + 36 = 0
   $$

4. **Solving the quadratic equation**:
   $$
   x = \frac{51 \pm \sqrt{51^2 - 4 \cdot 4 \cdot 36}}{2 \cdot 4} = \frac{51 \pm \sqrt{2025}}{8} = \frac{51 \pm 45}{8}
   $$
   
   This gives two solutions:
   - $$ x = \frac{51 + 45}{8} = \frac{96}{8} = 12 $$
   - $$ x = \frac{51 - 45}{8} = \frac{6}{8} = 0.75 $$

5. **Finding corresponding $$ y $$ values**:
   - If $$ x = 12 $$:
     $$
     y = 2(12) - 6 = 24 - 6 = 18
     $$
   - If $$ x = 0.75 $$:
     $$
     y = 2(0.75) - 6 = 1.5 - 6 = -4.5
     $$

**Solutions**:
- $$ x = 12 $$ and $$ y = 18 $$
- $$ x = 0.75 $$ and $$ y = -4.5 $$

---

### Problem 4.2: State the arithmetic and geometric sequences for each of your solutions.

**1. When $$ x = 12 $$ and $$ y = 18 $$:**

- **Arithmetic Progression (AP)**:
  $$
  6 ;\ 12 ;\ 18
  $$
  - Common difference $$ d = 6 $$ (since $$ 12 - 6 = 6 $$ and $$ 18 - 12 = 6 $$).

- **Geometric Progression (GP)**:
  $$
  12 ;\ 18 ;\ 27
  $$
  - Common ratio $$ r = 1.5 $$ (since $$ \frac{18}{12} = 1.5 $$ and $$ \frac{27}{18} = 1.5 $$).

**2. When $$ x = 0.75 $$ and $$ y = -4.5 $$:**

- **Arithmetic Progression (AP)**:
  $$
  6 ;\ 0.75 ;\ -4.5
  $$
  - Common difference $$ d = -5.25 $$ (since $$ 0.75 - 6 = -5.25 $$ and $$ -4.5 - 0.75 = -5.25 $$).

- **Geometric Progression (GP)**:
  $$
  0.75 ;\ -4.5 ;\ 27
  $$
  - Common ratio $$ r = -6 $$ (since $$ \frac{-4.5}{0.75} = -6 $$ and $$ \frac{27}{-4.5} = -6 $$).

---

**Summary of Solutions**:

1. **First Solution**:
   - $$ x = 12 $$, $$ y = 18 $$
   - **AP**: $$ 6 ;\ 12 ;\ 18 $$ with common difference $$ 6 $$
   - **GP**: $$ 12 ;\ 18 ;\ 27 $$ with common ratio $$ 1.5 $$

2. **Second Solution**:
   - $$ x = 0.75 $$, $$ y = -4.5 $$
   - **AP**: $$ 6 ;\ 0.75 ;\ -4.5 $$ with common difference $$ -5.25 $$
   - **GP**: $$ 0.75 ;\ -4.5 ;\ 27 $$ with common ratio $$ -6 $$
**Solutions:** 1. **First Solution**: - $$ x = 12 $$, $$ y = 18 $$ - **Arithmetic Sequence**: $$ 6, 12, 18 $$ with a common difference of $$ 6 $$ - **Geometric Sequence**: $$ 12, 18, 27 $$ with a common ratio of $$ 1.5 $$ 2. **Second Solution**: - $$ x = 0.75 $$, $$ y = -4.5 $$ - **Arithmetic Sequence**: $$ 6, 0.75, -4.5 $$ with a common difference of $$ -5.25 $$ - **Geometric Sequence**: $$ 0.75, -4.5, 27 $$ with a common ratio of $$ -6 $$

3. The first three terms of the sequence \( 6 ; x ; y ; 27 \) form an arithmetic progression and the last three terms form a geometric progression. 3.1 Determine the values of \( x \) and \( y \). 4.2 State the arithmetic and geometric sequences for each of your solutions.

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