1. Find the sum of the first 12 terms of the arithmetic series with the first term 10 and comman difference 8

To find the sum of the first 12 terms of an arithmetic series, you can use the formula \( S_n = \frac{n}{2} \times (2a + (n - 1)d) \), where \( S_n \) is the sum, \( n \) is the number of terms, \( a \) is the first term, and \( d \) is the common difference. Here, \( n = 12 \), \( a = 10 \), and \( d = 8 \). Plugging in the values: \[ S_{12} = \frac{12}{2} \times (2 \cdot 10 + (12 - 1) \cdot 8) = 6 \times (20 + 88) = 6 \times 108 = 648 \] So, the sum of the first 12 terms of the series is 648. Now, for a fun fact: Did you know that the concept of arithmetic series dates back to ancient civilizations? The ancient Greeks and Babylonians used these series for various mathematical calculations, showing that people have been fascinated with patterns in numbers for thousands of years! And speaking of patterns, when working with arithmetic series, a common mistake is miscalculating the number of terms or the common difference. Always double-check that you've got your first term and your difference right! Otherwise, your final sum could go a bit off course, just like trying to find your way without a map!