Pedagogical thoughts about sequences and series (Part 2)

After yesterday’s post about arithmetic and geometric sequences, I’d like to contribute some thoughts about teaching this topic, based on my own experience over the years.

1. Some students really resist the subscript notation a_n when encountering it for the first time. To allay these concerns, I usually ask my students, “Why can’t we just label the terms in the sequence as a, b, c, and so on?” They usually can answer: what if there are more than 26 terms? That’s the right answer, and so the a_n is used so that we’re not limited to just the letters of the English alphabet.

Another way of selling the a_n notation to students is by telling them that it’s completely analogous to the f(x) notation used more commonly in Algebra II and Precalculus. For a “regular” function f(x), the number x is chosen from the domain of real numbers. For a sequence a_n, the number n is chosen from the domain of positive (or nonnegative) integers.

2. The formulas in Part 1 of this series (pardon the pun) only apply to arithmetic and geometric sequences, respectively. In other words, if the sequence is neither arithmetic nor geometric, then the above formulas should not be used.

While this is easy to state, my observation is that some students panic a bit when working with sequences and tend to use these formulas on homework and test questions even when the sequence is specified to be something else besides these two types of sequences. For example, consider the following problem:

Find the 10th term of the sequence 1, 4, 9, 16, \dots

I’ve known pretty bright students who immediately saw that the first term was 1 and the difference between the first and second terms was 3, and so they answered that the tenth term is 1 + (10-1)\times 3 = 28… even though the sequence was never claimed to be arithmetic.

I’m guessing that these arithmetic and geometric sequences are emphasized so much in class that some students are conditioned to expect that every series is either arithmetic or geometric, forgetting (especially on tests) that there are sequences other than these two.

3. Regarding arithmetic sequences, sometimes it helps by giving students a visual picture by explicitly make the connection between the terms of an arithmetic sequence and the points of a line. For example, consider the arithmetic sequence which begins

13, 16, 19, 22, \dots

The first term is 13, the second term is 16, and so on. Now imagine plotting the points (1,13), (2,16), (3,19), and (4,22) on the coordinate plane. Clearly the points lie on a straight line. This is not surprising since there’s a common difference between terms. Moreover, the slope of the line is 3. This matches the common difference of the arithmetic sequence.

4. In ordinary English, the words sequence and series are virtually synonymous. For example, if someone says either, “a sequence of unusual events” or “a series of unusual events,” the speaker means pretty much the same thing

However, in mathematics, the words sequence and series have different meanings. In mathematics, an example of an arithmetic sequence are the terms

1, 3, 5, 7, 9, \dots, 99

However, an example of an arithmetic series would be

1 + 3 + 5 + 7 + 9 + \dots + 99

In other words, a sequence provides the individual terms, while a series is a sum of the terms.

When teaching this topic, I make sure to take a minute to emphasize that the words sequence and series will mean something different in my class, even though they basically mean the same thing in ordinary English.

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