Let's break down the problem step by step.

Problem Recap:

• We have an arithmetic sequence: $8, 2x, 2y$
• We have a geometric sequence: $2x, 2y, 36$

2.1.1 Show that $y = 2x - 4$ using the arithmetic sequence.

Step 1: Understand the condition of an arithmetic sequence.

In an arithmetic sequence, the difference between consecutive terms is constant. So, we can write: $2x - 8 = 2y - 2x$

Step 2: Simplify the equation.

Let's solve the equation: $2x - 8 = 2y - 2x$ Add $2x$ to both sides: $4x - 8 = 2y$ Divide the whole equation by 2: $2x - 4 = y$ Thus, we have shown that: $y = 2x - 4$

2.1.2 Determine the values of $x$ and $y$.

Step 1: Use the geometric sequence condition.

In a geometric sequence, the ratio between consecutive terms is constant. Therefore: $\frac{2y}{2x} = \frac{36}{2y}$ Simplifying this: $\frac{y}{x} = \frac{18}{y}$ Cross-multiply to get: $y^2 = 18x$

Step 2: Substitute the expression for $y$ from 2.1.1.

We know that $y = 2x - 4$. Substitute this into the equation: $(2x - 4)^2 = 18x$

Step 3: Expand and simplify the quadratic equation.

Expanding: $4x^2 - 16x + 16 = 18x$ Rearrange to form a standard quadratic equation: $4x^2 - 34x + 16 = 0$ Divide the entire equation by 2: $2x^2 - 17x + 8 = 0$

Step 4: Solve the quadratic equation.

Use the quadratic formula: $x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$ Here, $a = 2$, $b = -17$, and $c = 8$. Substituting these values: $x = \frac{17 \pm \sqrt{(-17)^2 - 4(2)(8)}}{4}$ $x = \frac{17 \pm \sqrt{289 - 64}}{4}$ $x = \frac{17 \pm \sqrt{225}}{4}$ $x = \frac{17 \pm 15}{4}$

So, the two possible values for $x$ are: $x = \frac{32}{4} = 8 \quad \text{or} \quad x = \frac{2}{4} = 0.5$

Step 5: Find the corresponding values of $y$.

If $x = 8$: $y = 2(8) - 4 = 16 - 4 = 12$

If $x = 0.5$: $y = 2(0.5) - 4 = 1 - 4 = -3$

Thus, the possible values for $x$ and $y$ are:

• $x = 8$, $y = 12$
• $x = 0.5$, $y = -3$

Conclusion:

The values of $x$ and $y$ can be $(8, 12)$ or $(0.5, -3)$.

Would you like more details or have any questions about this solution?

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1. What are the properties of arithmetic sequences?
2. How do you derive the general term of an arithmetic sequence?
3. What defines a geometric sequence, and how do you find its common ratio?
4. How can the quadratic formula be used to solve different types of quadratic equations?
5. In what other contexts might these types of sequences appear?

Tip: When working with sequences, always check if there's more than one possible solution, as seen here with both positive and negative values for $x$ and $y$.