2.3 The middle term of an arithmetic series of 31 terms is 50 . Detcrmine the sum of the series.

In an arithmetic series, the middle term can be found using the formula for the term in the middle position. Since there are 31 terms, the middle term is the 16th term. If the 16th term is 50, then the first term (a) and the common difference (d) can be used to express it as \( a + 15d = 50 \), establishing a relationship between them. Next, the sum of the series can be calculated using the formula for the sum of an arithmetic series: \( S_n = \frac{n}{2} (a + l) \), where \( n \) is the number of terms, and \( l \) is the last term. To find the last term, the 31st term would be \( a + 30d \). However, since the sum can also be expressed using the middle term, we have: \[ S = n \times \text{Middle term} = 31 \times 50 = 1550. \] Therefore, the sum of the series is **1550**.