If \( 8 ; 2 x ; 2 y \) forms an arithmetic sequence and \( 2 x ; 2 y ; 36 \) forms a geometric sequence determine the values of \( x \) and \( y \).

To solve for \(x\) and \(y\), let's start with the conditions given by the arithmetic sequence and the geometric sequence. 1. For the arithmetic sequence \(8; 2x; 2y\): The middle term is the average of the other two terms, so we set up the equation: \[ 2x = \frac{8 + 2y}{2} \] Multiplying both sides by 2 to eliminate the fraction: \[ 4x = 8 + 2y \quad (1) \] 2. For the geometric sequence \(2x; 2y; 36\): The square of the middle term is equal to the product of the other two terms, so we have: \[ (2y)^2 = 2x \cdot 36 \] Simplifying this gives: \[ 4y^2 = 72x \quad (2) \] Now we can solve the system of equations. From equation (1), rearranging gives: \[ 2y = 4x - 8 \quad \Rightarrow \quad y = 2x - 4 \quad (3) \] Next, we substitute equation (3) into equation (2): \[ 4(2x - 4)^2 = 72x \] Expanding the left side: \[ 4(4x^2 - 16x + 16) = 72x \] This simplifies to: \[ 16x^2 - 64x + 64 = 72x \] Combining like terms: \[ 16x^2 - 136x + 64 = 0 \] Now, we can divide the entire equation by 4: \[ 4x^2 - 34x + 16 = 0 \] Using the quadratic formula where \(a = 4\), \(b = -34\), and \(c = 16\): \[ x = \frac{-(-34) \pm \sqrt{(-34)^2 - 4 \cdot 4 \cdot 16}}{2 \cdot 4} \] Calculating the discriminant: \[ (-34)^2 - 4 \cdot 4 \cdot 16 = 1156 - 256 = 900 \] Taking the square root gives: \[ \sqrt{900} = 30 \] Thus, substituting back into the quadratic formula: \[ x = \frac{34 \pm 30}{8} \] This results in two possible values for \(x\): 1. \(x = \frac{64}{8} = 8\) 2. \(x = \frac{4}{8} = \frac{1}{2}\) Now we find corresponding values for \(y\) using equation (3): 1. If \(x = 8\): \[ y = 2(8) - 4 = 16 - 4 = 12 \] 2. If \(x = \frac{1}{2}\): \[ y = 2\left(\frac{1}{2}\right) - 4 = 1 - 4 = -3 \] Hence, the pairs \((x, y)\) can be either \((8, 12)\) or \(\left(\frac{1}{2}, -3\right)\). The final answers are: \[ (x, y) = (8, 12) \quad \text{or} \quad \left(\frac{1}{2}, -3\right) \]