geometric sequence – Page 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.

I’m not particularly a fan of memorizing formulas. Apparently, most college students aren’t fans either, because they often don’t have immediate recall of certain formulas from high school when they’re needed in the collegiate curriculum.

While I’m not a fan of making students memorize formulas, I am a fan of teaching students how to derive formulas. Speaking for myself, if I ever need to use a formula that I know exists but have long since forgotten, the ability to derive the formula allows me to get it again.

Which leads me to today’s post: the derivation of the formulas for the $n$ th term of an arithmetic sequence and of a geometric sequence. This topic is commonly taught in Precalculus but, in my experience, is often forgotten by students years later when needed in later classes.

An arithmetic sequence is specified by two numbers: the first term and the common difference between terms. For example, if the first term is $16$ and the common difference is $3$ , then the sequence begins as

$16, 19, 22, 25, 28, 31, 34, \dots$

If the first term is $29$ and the common difference is $-4$ , then the sequence begins as

$29, 25, 21, 17, 13, 9, 5, 1, -3, \dots$

For those of us old enough to remember, our favorite arithmetic sequences came from Schoolhouse Rock:

Let’s discuss the first arithmetic sequence, whose first seven terms are:

$16, 19, 22, 25, 28, 31, 34, \dots$

How do we get the $8$ th term? That’s easy: we just add $3$ to $34$ to get $37$ .

How to we get the $100$ th term. That’s easy: we just add $3$ to the $99$ th term.

Oops. We don’t know the $99$ th term. To get the $99th$ term, we need the $98$ th term, which in turn requires the $97$ th term. Et cetera, et cetera, et cetera.

The trouble (so far) is that an arithmetic sequence is recursively defined: to get one term, I add something to the previous term. Mathematically, the arithmetic sequence is defined by

$a_n = a_{n-1} + d$ ,

where $d$ is the common difference. This can be very intimidating to students when seeing it for the first time. So, to make this formula less intimidating, I usually read this equation as “Each next term in the sequence is equal to the previous term in the sequence plus the common difference.”

It would be far better to have a closed-form formula, where I could just plug in $100$ to get the $100$ th term, without first figuring out the previous $99$ terms.

To this end, we notice the following pattern:

Second term: $19 = 16 + 3$
Third term: $22 = 19 + 3 = 16 + 3 + 3 = 16 + 2 \times 3$
Fourth term: $25 = 22+ 3 = 16 + (2 \times 3) + 3 = 16 + 3 \times 3$
Fifth term: $28 = 25+ 3 = 16 + (3 \times 3) + 3 = 16 + 4 \times 3$
Sixth term: $31 = 28+ 3 = 16+ (4 \times 3) + 3 = 16 + 5 \times 3$
Seventh term: $34 = 31 + 3 = 16 + (5 \times 3) + 3 = 16 + 6 \times 3$

It looks like we have a pattern, so we can guess that:

One hundredth term = $16 + (100-1) \times 3 = 313$

In general, we have justified the closed-form formula

$a_n = a_1 + (n-1)d$ ,

where $a_1$ is the first term, and $d$ is the common difference. In words: to get the $n$ th term of an arithmetic sequence, we add $d$ to the first term $n-1$ times. (This may be formally proven using mathematical induction, though I won’t do so here.)

A closed-form formula for a geometric sequence is similarly obtained. In a geometric sequence, each term is equal to the previous term multiplied by a common ratio. Mathematically, the geometric sequence is recursively defined by

$a_n = a_{n-1}r$ ,

where $r$ is the common ratio. For example, if the first term is $3$ and the common ratio is $2$ , then the first few terms of the sequence are

$3, 6, 12, 24, 48, dots$

By the same logic used above, to get the $n$ th term of an geometric sequence, we multiply $r$ to the first term $n-1$ times. Thus justifies the formula

$a_n = a_1 r^{n-1}$ ,

which may be formally proven using mathematical induction.

Tag: geometric sequence

Pedagogical thoughts about sequences and series (Part 2)

Formulas for arithmetic and geometric sequences (Part 1)