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

Let's denote the first three terms of the arithmetic progression (AP) as \( a \), \( a + d \), and \( a + 2d \), where \( a \) is the first term and \( d \) is the common difference. 1. **Sum of the first three terms:** \[ a + (a + d) + (a + 2d) = 3a + 3d = 27 \] Simplifying: \[ a + d = 9 \quad \text{(Equation 1)} \] 2. **Product of the first two terms:** \[ a \times (a + d) = 36 \] Using Equation 1 (\( a + d = 9 \)): \[ a \times 9 = 36 \implies a = 4 \] 3. **Finding the common difference \( d \):** Substitute \( a = 4 \) into Equation 1: \[ 4 + d = 9 \implies d = 5 \] 4. **First three terms of the AP:** \[ \begin{align*} \text{First term} &= a = 4, \\ \text{Second term} &= a + d = 4 + 5 = 9, \\ \text{Third term} &= a + 2d = 4 + 10 = 14. \end{align*} \] **Verification:** - Sum: \( 4 + 9 + 14 = 27 \) - Product: \( 4 \times 9 = 36 \) Thus, the first three terms of the progression are **4, 9, and 14**. **Answer:** The first three terms are 4, 9, and 14.