\( x ; y ; 8 ; \ldots \) is a geometric sequence and \( 1 ; x+1 ; y+1 ; \ldots \) is an arithmetic sequence. Determine the values of \( x \) and \( y \).

In a geometric sequence, the ratio between consecutive terms is constant. Here, the second term divided by the first term must equal the third term divided by the second term: \[ \frac{y}{x} = \frac{8}{y}. \] Cross-multiplying gives: \[ y^2 = 8x \quad (1). \] In an arithmetic sequence, the difference between consecutive terms is constant. Here, the second term minus the first term must equal the third term minus the second term: \[ (x+1) - 1 = (y+1) - (x+1). \] Simplifying this yields: \[ x = y - x \quad \Rightarrow \quad 2x = y \quad (2). \] Now we can substitute equation (2) into equation (1): \[ (2x)^2 = 8x. \] This simplifies to: \[ 4x^2 = 8x \quad \Rightarrow \quad 4x^2 - 8x = 0. \] Factoring out \( 4x \): \[ 4x(x - 2) = 0. \] Thus, the solutions are \( x = 0 \) or \( x = 2 \). 1. If \( x = 0 \): - Using equation (2), \( y = 2(0) = 0 \). This gives us the values \( (x, y) = (0, 0) \). 2. If \( x = 2 \): - Using equation (2), \( y = 2(2) = 4 \). This gives us the values \( (x, y) = (2, 4) \). To verify which pair satisfies the sequences: - For \( (0, 0) \): - The geometric sequence is \( 0, 0, 8, \ldots \) (not a valid sequence). - For \( (2, 4) \): - The geometric sequence is \( 2, 4, 8, \ldots \) with a common ratio of 2. - The arithmetic sequence is \( 1, 3, 5, \ldots \) with a common difference of 2. Thus, the values of \( x \) and \( y \) are: \[ \boxed{(2, 4)}. \]