\( \sum _ { i = 0 } ^ { m } ( 1 - 3 i ) = - 671 \)

To understand the series \( \sum_{i=0}^{m} (1 - 3i) \), we can rewrite it as \( \sum_{i=0}^{m} 1 - \sum_{i=0}^{m} 3i \). The first part, \( \sum_{i=0}^{m} 1 \), is simply \( m + 1 \). The second part is an arithmetic series—the sum of the first \( m \) integers multiplied by 3, which equals \( 3\frac{m(m+1)}{2} \). Putting it all together: \[ m + 1 - \frac{3m(m+1)}{2} = -671. \] Now, if we expand and solve for \( m \), we can find the value of \( m \). It’s essential to turn the equation into a standard polynomial format to deduce \( m \). --- The study of arithmetic series like this has a rich history dating back to ancient mathematicians such as Gauss, who famously summed integers from 1 to 100 in a matter of seconds by noticing pairs. Today, this concept underpins algebra and calculus, helping us problem-solve in fields as varied as economics and physics. In practice, understanding summations is crucial in programming, especially in algorithms where you calculate cumulative values. You might encounter instances in data analysis where you need to quickly sum arrays or sequences, making this mathematical foundation incredibly useful in creating efficient codes and understanding algorithm complexities.