How Much Mathematics Should A Student Memorize? Part 4, Geometric Series

In teaching mathematics for many years, one of the things I emphasized over and over again was that students should memorize the absolute minimum necessary, and then I did my best to make explicit what this absolute minimum is. It is better, I explained, to spend time solving problems, discussing applications, “reading around the subject,” and so on, rather than waste time on sterile memorization.

And certainly the best way to memorize something in mathematics is to use it repeatedly, meaningfully. After teaching a subject repeatedly, one naturally memorizes things.

Having said all this, I have to admit that sometimes I lied in order to drive home this point. That is, I sometimes pretended that I did not remember certain things, in order to work them out on the spot, in an attempt to illustrate the point that it was not necessary to have memorized them in the first place. Sorry!

But there is one formula that I have real problems remembering, and so I really do have to work it out from scratch every time I want to use it (looking it up is too much of a hassle, and it is really fast to work out, as I’ll show you in  a moment). It’s the formula for the sum of a finite geometric series.

The trick to work out the formula really is easy to remember, and it works for both finite and infinite geometric series. And this includes repeating decimals, which are a kind of infinite geometric series.

First, to get the formula for the sum of an infinite geometric series, call the sum S:

S = 1 + r + r^2 + r^3 + \ldots

Now multiply both sides of the previous equation by r:

rS = r + r^2 + r^3 + r^4 + \ldots

Subtract the second equation from the first, noticing that all of the terms except one cancel on the right side:

S - rS = 1

which means that

S = \dfrac{1}{1 - r}

Note that this trick is just a memory aid, and the verification that the formula really is valid (for -1 < r < 1) requires a certain amount of formal justification. But the nice thing is that the very same trick works for finite geometric series:

S = 1 + r + r^2 + r^3 + \ldots + r^n

Now multiply both sides of the previous equation by r:

rS = r + r^2 + r^3 + r^4 + \ldots + r^n + r^{n+1}

Subtract the second equation from the first, noticing again the nice cancellation on the right side:

S - rS = 1 - r^{n+1}

Solving for S gives the desired formula:

S(1 - r) = 1 - r^{n+1}

S = \dfrac{1 - r^{n+1}}{1 - r}

Full disclosure: I remember the structure of this formula, but I don’t have confidence in the exponent of r in the numerator. Is it n or is it (n + 1)? I’m never sure, so I have to work it out from scratch each time. I speculate that the reason for this is that I’ve seen the formula too many times in too many books in more than one form. Some books stop at r^{n-1} instead of r^n in the setup, and so they end up with a different formula. I’m too lazy to try to sort this through (but really, how hard is it? Not very …), so I just work it out every time. Thankfully, it only takes seconds, which justifies (don’t you agree?) my not bothering to carefully remember the formula.

Delightfully, the same trick allows one to easily convert a repeating decimal to a fraction. For example, consider the repeating decimal 0.34343434343434 … . You can see that this is just a geometric series, right? Then the same trick should work. Call the decimal S, then multiply both sides of the resulting equation by 100 (why 100?), and then subtract and solve for S:

S = 0.34343434 \ldots

100S = 34.34343434 \ldots

100S - S = 34

99S = 34

S = \dfrac{34}{99}

Voila!

ps. Back to my exhortation to students to “memorize the absolute minimum.” Sometimes in class I would solve problems in class in two ways; one relying on the memorization of a formula (usually a standard formula highlighted in the course textbook), and the other without using the formula. After presenting both solutions, I would poll the class and ask which solution was preferred, and almost always the majority of the class preferred the solution that relied on the memorization of a formula. This is part of human nature, I believe: thinking is hard. It is easier for many people to memorize formulas (which are no doubt forgotten soon after the final exam) than to truly understand the subject. But this is not the fault of our students, but rather the fault of the structure of our school system, which relies on high-stakes testing (much of it which emphasized technique over deep understanding), heavily penalizes mistakes, and rushes students through courses at such a rapid pace that very few have much chance of truly understanding a course well before it ends. (I’ve discussed aspects of this problem a number of times, including yesterday.) A lucky few (primarily teachers!) have the opportunity to go over the material again some day, and then their understanding grows.

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About Santo D'Agostino

I am a former university prof, having taught mathematics and physics since the mid 1980s. I have also been a textbook writer/editor since then. Currently I am working independently on a number of writing and education projects. I love math and physics, and love teaching and writing about them. My blog also discusses education, science, environment, etc. http://qedinsight.wordpress.com Further resources, and online tutoring, can be found at my other site http://www.qedinfinity.com
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2 Responses to How Much Mathematics Should A Student Memorize? Part 4, Geometric Series

  1. Mathew Menonkariyil says:

    Great post professor,

    I have a slightly different views on the reason why students on average prefer the rote memorization approach.It is because they see Mathematics as something rather dull and merely a credit that needs to be obtained and as such they’re interested in the bare minimum.There isn’t much emphasis on Mathematics as the language of the universe and that nature obeys the very laws of Mathematics we write down as equations on a piece of paper or there isn’t much of a stress on creating a sense of awe and wonder of Mathematics.The extent to which students eschew Math and science is the extend to which they isolate themselves from reality.I guess there is this implicit assumption that reality is one thing while Math and Science is something else ,perhaps driving a wedge between this false dichotomy can help quite a bit.If this is stressed on the tests can get as hard as they can get and students will still manage to do quite well.I reason as such because I observe this with students who claim they can’t sit still and practice Mathematics, the very same students will spend hours sitting still playing a computer game or watching television how can an allegedly rambunctious student claim to be unable to focus their mind when it comes to Math but can do so easily with games and tv ?I believe it is because games and tv captures their imagination to a much larger extent.Unfortunately, I think this is too idealistic there are some students who will never do Mathematics for the sake of doing it or because its fun .Another point worth making is that Mathematics is ever more relevant in the job market today and the economies of the 21st century are Math based, being Mathematically literate can give an edge for potential job seekers.

    • Hi Matthew,

      Thanks for the thoughtful reply. I think you’ve hit on a very important point, that many students don’t see mathematics as intimately connected with nature. Part of the blame for that has to go to the education system; the way we have overloaded each course with so much content, and the way we have structured our assessments means that there is a lot of pressure to place various calculational procedures into short-term memory (to pass the exams), and very little time to experience the sense of awe and wonder that comes with a deep exploration of mathematics in nature.

      Your second point reinforces the first one: If only we could find a way to modify our education system appropriately, then we could better encourage students to devote the time needed to develop a much deeper appreciation of the mathematical perspective on nature. Children are naturally curious, but years of grubbing for check marks and grades effectively pounds the curiousity out of most of us. A few lucky ones survive the ordeal with their enthusiasm intact, but surely we can do better than this.

      Your final point about the job market is also a strong one. The irony is that in our drive to prepare students for technical careers, we overload them with content, when we should instead be helping them to develop the qualities that are truly essential: imagination, creativity, critical thinking, problem solving, modelling. But content is so much easier to assess, and so much easier to convey. Alas.

      Thanks again for the thoughtful reply, Matthew!

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