To solve the problem, we need to find the common difference of an arithmetic progression (A.P.) given the 3rd and 9th terms. Let's break this down step-by-step. ### Step 1: Understanding the Terms of an A.P. In an arithmetic progression, the \( n \)-th term can be expressed as: \[ a_n = a + (n-1)d \] where: - \( a \) is the first term, - \( d \) is the common difference, - \( n \) is the term number. ### Step 2: Setting Up the Equations From the problem, we know: - The 3rd term \( a_3 = 4x - 2y \) - The 9th term \( a_9 = 10x - 8y \) Using the formula for the \( n \)-th term, we can write: 1. For the 3rd term: \[ a_3 = a + 2d = 4x - 2y \quad \text{(1)} \] 2. For the 9th term: \[ a_9 = a + 8d = 10x - 8y \quad \text{(2)} \] ### Step 3: Setting Up a System of Equations Now we have two equations: 1. \( a + 2d = 4x - 2y \) (Equation 1) 2. \( a + 8d = 10x - 8y \) (Equation 2) ### Step 4: Eliminating \( a \) To find \( d \), we can eliminate \( a \) by subtracting Equation 1 from Equation 2: \[ (a + 8d) - (a + 2d) = (10x - 8y) - (4x - 2y) \] This simplifies to: \[ 6d = 10x - 8y - 4x + 2y \] \[ 6d = 6x - 6y \] Now, we can simplify this further: \[ d = \frac{6x - 6y}{6} = x - y \] ### Step 5: Conclusion Thus, the common difference \( d \) is: \[ d = x - y \] ### Step 6: Evaluating the Options Now, let's evaluate the options provided: - **A. 19x - 17y**: This does not match our result. - **B. 8x - 4y**: This does not match our result. - **C. x - y**: This matches our result. - **D. 2x**: This does not match our result. ### Final Answer The correct option is **C. x - y**. ### Summary of Key Points - The \( n \)-th term of an A.P. is given by \( a + (n-1)d \). - We set up equations for the 3rd and 9th terms and eliminated \( a \) to find \( d \). - The common difference \( d \) was found to be \( x - y \). - Always check each option against your derived answer to confirm correctness.