1.6 The sum of the first 15 terms of an arithmetic progression is 165 . The common difference is 2 . Calculate the first term of the sequence.

To find the first term of the arithmetic progression (AP), we can use the formula for the sum of the first \( n \) terms of an AP: \[ S_n = \frac{n}{2} \times (2a + (n-1)d) \] where: - \( S_n \) is the sum of the first \( n \) terms, - \( n \) is the number of terms, - \( a \) is the first term, - \( d \) is the common difference. Given: - \( S_{15} = 165 \) - \( n = 15 \) - \( d = 2 \) We can substitute these values into the formula: \[ 165 = \frac{15}{2} \times (2a + (15-1) \cdot 2) \] Simplifying the equation: \[ 165 = \frac{15}{2} \times (2a + 28) \] Now, multiply both sides by 2 to eliminate the fraction: \[ 330 = 15 \times (2a + 28) \] Next, divide both sides by 15: \[ 22 = 2a + 28 \] Now, isolate \( 2a \): \[ 2a = 22 - 28 \] \[ 2a = -6 \] Now, divide by 2 to find \( a \): \[ a = -3 \] Thus, the first term of the sequence is: \[ \boxed{-3} \]