15. The sum of the first 3 consecutive terms of an arithmetic progressi is 27 . The product of the first 2 terms is 36 . What are the first 3 ter of the progression?

Let's solve the problem step by step. **Given:** - The first three terms of an arithmetic progression (AP) are \( a \), \( a + d \), and \( a + 2d \), where \( a \) is the first term and \( d \) is the common difference. - The sum of the first three terms is 27. - The product of the first two terms is 36. **Step 1: Express the Sum of the First Three Terms** \[ a + (a + d) + (a + 2d) = 3a + 3d = 27 \] \[ 3(a + d) = 27 \] \[ a + d = 9 \quad \text{(Equation 1)} \] **Step 2: Express the Product of the First Two Terms** \[ a \times (a + d) = 36 \] From Equation 1, substitute \( a + d = 9 \): \[ a \times 9 = 36 \] \[ a = 4 \] **Step 3: Find the Common Difference \( d \)** Using Equation 1: \[ 4 + d = 9 \] \[ d = 5 \] **Step 4: Determine the First Three Terms** \[ \text{First term} = a = 4 \] \[ \text{Second term} = a + d = 4 + 5 = 9 \] \[ \text{Third term} = a + 2d = 4 + 2(5) = 14 \] **Verification:** - **Sum:** \( 4 + 9 + 14 = 27 \) ✓ - **Product:** \( 4 \times 9 = 36 \) ✓ **Answer:** The first three terms of the arithmetic progression are 4, 9, and 14.