Exponential growth and decay (Part 5): Paying off credit-card debt via recurrence relations

The following problem in differential equations has a very practical application for anyone who has either (1) taken out a loan to buy a house or a car or (2) is trying to pay off credit card debt. To my surprise, most math majors haven’t thought through the obvious applications of exponential functions as a means of engaging their future students, even though it is directly pertinent to their lives (both the students’ and the teachers’).

You have a balance of $2,000 on your credit card. Interest is compounded continuously with a rate of growth of 25% per year. If you pay the minimum amount of $50 per month (or $600 per year), how long will it take for the balance to be paid?

In previous posts, I approached this problem using differential equations. There’s another way to approach this problem that avoids using calculus that, hypothetically, is within the grasp of talented Precalculus students. Instead of treating this problem as a differential equation, we instead treat it as a first-order difference equation (also called a recurrence relation):

A_{n+1} = r A_n - k

The idea is that the amount owed is multiplied by a factor r (which is greater than 1), and from this product the amount paid is deducted. With this approach — and unlike the approach using calculus — the payment period would be each month and not per year. Therefore, we can write

A_{n+1} = \displaystyle \left( 1 + \frac{0.25}{12} \right) A_n - 50

Notice that the meaning of the 25% has changed somewhat… it’s no longer the relative rate of growth, as the 25% has been equally divided for the 12 months.

green lineA full treatment of the solution of difference equations belongs to a proper course in discrete mathematics. However, this particular difference equation can be solved in a straightforward fashion that should be accessible to talented Precalculus students. Let’s use the above recurrence relation to try to find a pattern. For n = 1, we find

A_1 = r A_0 - k = r P - k.

For n = 2, we find

A_2 = r A_1 - k

A_2 = r (rP - k) - k

A_2 = r^2P - rk - k

A_2 = r^2 P - k (1 + r)

For n = 3, we find

A_3 = r A_2 - k

A_3 = r \left[ r^2 P - k(1+r) \right] - k

A_3 = r^3 - rk(1+r) - k

A_3 = r^3 P - rk - r^2k - k

A_3 = r^2 P - k \left( 1 + r+r^2 \right)

At this point, we can probably guess a pattern:

A_n = r^n P - k \left( 1 + r + r^2 + \dots + r^{n-1} \right)

Using the formula for a finite geometric series, this simplifies as

A_n = r^n P - k \left( \displaystyle \frac{1 - r^n}{1-r} \right).

Indeed, though I won’t do it here, this can be formally proven using mathematical induction.


Engaging students: Arithmetic sequences

In my capstone class for future secondary math teachers, I ask my students to come up with ideas for engaging their students with different topics in the secondary mathematics curriculum. In other words, the point of the assignment was not to devise a full-blown lesson plan on this topic. Instead, I asked my students to think about three different ways of getting their students interested in the topic in the first place.

I plan to share some of the best of these ideas on this blog (after asking my students’ permission, of course).

This student submission comes from my former student Erick Cordero. His topic, from Precalculus: arithmetic sequences.

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“What interesting word problems using this topic can your students do now?”


There are many word problems we can do with arithmetic sequences but I am going to give one example that I believe students will understand. For this example, lets suppose that John Q, a pre-calculus student, has just bought a new phone from apple, but because of this new upgrade, Q’s parents are concern with the sum of money they will be paying for his monthly bill. Q’s first bill happens to be $65, his total after the second bill is $130, after the third bill the running sum is $195, if this pattern continues, how many months will it take for the total to reach $780? To solve this problem we would write the terms in a sequence starting with the first term being $65 and up to three more terms. After writing out a few terms, I would expect the students to find the common difference between the terms and then compute the slope of the terms (I say slope because I hope they can see that this pattern is linear and therefore we can model the data using a linear equation and not just use the formula for arithmetic sequence but rather derive one ourselves). Then just like the students did in algebra one, they can use the point slope formula to come up with an equation for the sequence. I would explain to the students that now that we have the formula we can easily find the nth term that contains our sum, and this parallels the same process as having an x value and finding a corresponding y value and by using this process I can assure the students that the methods they learned in algebra are still important in pre-calculus.



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“How can this topic be used in your students’ future courses in mathematics?”

Sequences and equations is a very important topic in mathematics, and unfortunately many students that take pre-calculus in high school will never get to experience how sequences evolve from simple arithmetic sequences to the more powerful ones in calculus II. Sequences are often overlook by students in pre-calculus (high school) because it is different from what they have encountered in their math career thus far, but maybe if we show students how this topic evolves in calculus II then they will pay more attention to it (Or they will forget it more since many students will not take calculus II). But from an educators’ standpoint, we understand how important sequences are. In calculus II teachers teach students how the elementary ideas they learned in pre-calculus are now used in calculus applications. One of these ideas is called a power series. Power series are fundamental to the study of calculus because they provide a way to represent some of the most important functions in our field. Power series are also useful in physics and chemistry. We also have Taylor Series, which have been regarded by some as the most interesting topic in calculus II. It is here, in calculus II where we see the true power of sequences and for some of us, that random topic in pre-calculus about sequences starts to make sense. Sequences is a topic that in rooted deep in the heart of mathematics and we should tell our students in pre-Cal, or algebra, how important this topic is as they go deeper into their math or science careers.

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“How can technology be used to effectively engage students with this topic?”


One website that I have often visit is Khan Academy, and I would encourage my students to do the same. I like this website because unlike some of the YouTube videos, these videos are more engaging and interesting. The person doing the videos is also more professional and has an understanding of mathematics beyond some of the YouTube clips I have encountered. The quality of this website is the best I have seen. I also like how Sal Khan (the person doing the videos) uses a lot of colors in his videos because it helps the students distinguish information. This is another reason why YouTube is sometimes not a great idea. Some of the videos are of people solving math problems on a white board, if that’s the point then why show the clip in the first place? Students do not want to see that, I will do enough of that. I have said enough bad things about YouTube, and hence it is only fair that I now show something positive from it.

The above is a YouTube clip from Khan Academy where Khan does a problem trying to find the 100th term of a sequence. Khan Academy is great place were students can see more examples of certain classroom topics but of course this is not something to replace classroom work but rather another option to engage students with.


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.

Formulas for arithmetic and geometric sequences (Part 1)

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 nth 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.

green lineAn 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 8th term? That’s easy: we just add 3 to 34 to get 37.

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

Oops. We don’t know the 99th term. To get the 99th term, we need the 98th term, which in turn requires the 97th 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 100th 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 nth 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.)

green lineA 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 nth 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.