Determine the arithmetic series if the sum of the first 10 terms is 185 and the tenth term is 5 .

The sum of the first 10 terms of an arithmetic series can be represented by the formula \( S_n = \frac{n}{2} (a + l) \), where \( n \) is the number of terms, \( a \) is the first term, and \( l \) is the last term. Given that \( S_{10} = 185 \) and \( l = a + 9d \) (where \( d \) is the common difference), you can set up the equations to solve for \( a \) and \( d \). You can determine that \( a = 5 - 9d \) by substituting known values and then find \( d \) using the sum formula. In real-world applications, arithmetic series can help predict future trends or aggregate data. For example, if you’re saving a fixed amount of money each month, knowing how to calculate the total savings over a specific number of months uses the principles of arithmetic series, helping you plan your finances more effectively and budget for larger expenses!