The first term of an arithmetic sequence is 54 , the last term is 19 and the sum of the terms of the sequence is 219 . Find the number of terms and the common difference.

To solve this problem, we can use the formulas for an arithmetic sequence. Let \( n \) be the number of terms and \( d \) the common difference. The sum of an arithmetic sequence can be given by the formula: \[ S_n = \frac{n}{2} \times (a + l) \] where \( S_n \) is the sum, \( a \) is the first term, and \( l \) is the last term. Here, \( a = 54 \), \( l = 19 \), and \( S_n = 219 \). Plugging these in, we get: \[ 219 = \frac{n}{2} \times (54 + 19) \] \[ 219 = \frac{n}{2} \times 73 \] \[ 438 = n \times 73 \] \[ n = \frac{438}{73} = 6 \] Now, knowing that the last term \( l = a + (n-1)d \), we can find \( d \): \[ 19 = 54 + (6 - 1)d \] \[ 19 = 54 + 5d \] \[ 5d = 19 - 54 \] \[ 5d = -35 \] \[ d = -7 \] So, the number of terms \( n \) is 6 and the common difference \( d \) is -7.