On his first day as a telemarketer, Marshall made 24 calls. His goal was to make 5 more calls on each successive day than he had made the day before. If Marshall met, but did not exceed, his goal, how many calls had he made in all after spending exactly 20 days making calls as a telemarketer? A. 670 B. 690 C. \(\quad 974\) D. \(1,430\) E. \(1,530\)

When we talk about arithmetic progression (AP), we are referring to a sequence of numbers in which the difference between any two consecutive terms is constant. This difference is known as the common difference, denoted as 'd'. An example of an arithmetic progression is the sequence 2, 4, 6, 8, 10,... where the common difference is 2.

In the context of the given exercise, Marshall's call pattern forms an arithmetic progression because each day he makes 5 more calls than the previous day, making the common difference 5. The sequence begins with 24 calls on the first day, 29 on the second, and so forth. Understanding this concept is crucial because it provides the structure we need to calculate the sum of calls made over a certain period.

The sum of an arithmetic series is the total of all the terms in an arithmetic progression. To calculate this sum, we have a specific formula:

\[ S = \frac{n}{2}(2a + (n-1)d) \]

where 'S' is the sum of the series, 'n' is the number of terms, 'a' represents the first term, and 'd' is the common difference. The formula combines these elements to give us the total sum in a more efficient way than adding each term individually.

This formula is a shortcut to what could be a lengthy addition process, especially with a large number of terms. For instance, in Marshall's situation, using the formula allowed us to quickly determine the total number of calls he made over the 20 days without having to list and sum all 20 numbers in the series.

The terms sequence and series are often used interchangeably, but they have distinct meanings in mathematics. A sequence is an ordered list of numbers following a particular pattern, while a series is the sum of the elements of a sequence.

• An arithmetic sequence is a list of numbers with a constant difference between consecutive terms (like the number of calls Marshall makes each day).
• A series is what you get when you add up the terms of a sequence. If we sum the number of calls from all 20 days, we get an arithmetic series.

In essence, a sequence describes the pattern or rule that generates the numbers, whereas a series is concerned with the total or accumulated value when these numbers are combined.