As we’ve discussed, the formula for an arithmetic series is

,

where is the number of terms, is the first term, is the common difference, and is the last term. This formula may be more formally expressed as

For homework and on tests, students are asked to directly plug into this formula and to apply this problem with word problems, like finding the total number of seats in an auditorium with 50 rows, where there are 12 seats in the front row and each row has two more seats than the row in front of it.

In my opinion, the ability to solve questions like the one below is the acid test for determining whether a student — who I assume can solve routine word problems like the one above — really understands series or is just familiar with series. In other words, if a student can solve routine word problems but is unable to handle a problem like the one below, then there’s still room for that student’s knowledge of series to deepen.

Calculate

There are two reasonable approaches for solving this problem.

Solution #1. Notice that . So this is really an arithmetic series whose first term is and whose common difference is . Therefore,

However, I’m supposed to start the series on , not . That means that I need to subtract off the first ten terms of the above series. Now

Finally,

Solution #2. Writing out the terms, we see that

or

The right-hand side is an arithmetic series whose “first” term is and whose last (th) term is . Therefore,

Of the two solutions, I suppose I have a mild preference for the first, as the second solution won’t work for something like . However, both solution demonstrate that the student is actually thinking about the meaning of the series instead of just plugging numbers in a formula, and so I’d be happy with either one in a Precalculus class.

I'm a Professor of Mathematics and a University Distinguished Teaching Professor at the University of North Texas. For eight years, I was co-director of Teach North Texas, UNT's program for preparing secondary teachers of mathematics and science.
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