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 Danielle Pope. Her topic, from Precalculus: arithmetic sequences.

green lineHow can this topic be used in your students’ future courses in mathematics or science?

In the future, the topic of arithmetic sequences will be built upon by introducing another sequence, the geometric sequence. A geometric sequence is just a sequence of multiples instead of increasing by a constant. The next topic introduced will be finding the sum of a sequence of numbers. This will be introduced as a series. The summation symbol will also be introduced to kids and they will learn that new notation. Summations will bring along many formulas for finding the leading coefficient and will show up later in Calculus 2 classes when talking about convergence and divergence of series. Another one of the things that kids will always be doing with sequences and series is finding the general form of a given sequence or series. Through school, this idea will never change the sequence and series will just get harder to identify.
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What interesting things can you say about the people who contributed to the discovery and/or the development of this topic? (You might want to consult Math Through The Ages.)

An arithmetic sequence is a set of numbers that have a constant difference between each term. One of the main people that come up when researching these sequences is Carl Friedrich Gauss. Many math-loving people know him as the “Prince of mathematicians”. He is famous for coming up with the equations to solve the sum of an arithmetic sequence. This comes as no surprise that he came up with this formula. The surprising thing about this realization is that he made it at an age young enough to still be in grade school. Stories say that Gauss was asked to solve for the sum on the board in grade school and used the formula of M ( M + 1 ) / 2 to solve for the correct answer. This just goes to show that anyone can, in fact, contribute to the greater good of mathematics at any age.

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How have different cultures throughout time used this topic in their society?

One of the first civilizations that utilized sequences was the Egyptians. They used the sequence of multiples of 2 to do their multiplication. The basic sequence is 1, 2, 4, 8, 16, 32, … and we are trying to solve 24 x 13 with the process pictured below.

The process behind this is to write the multiple of 2 sequences down the left side of the paper until you reach the largest multiple of 2 without going over the second number being multiplied, in this case, 13. Once that is done set the first term on the right side equal to the first number being multiplied, in this case, 24. Next, multiply the right side by 4 until you get the same amount of terms on the left side. Lastly find the sum of numbers on the left that add to 13, which are 1, 4, and 8. Add the corresponding multiples from the side, 24 + 96 + 192 = 312. The right side sum of the corresponding numbers checked on the left gives the product of the original problem, i.e. 312. This trick is cool to show just on its own but it’s also cool because it uses something as simple as a specific list of numbers aka a sequence of numbers.

References

http://www.softschools.com/facts/scientists/carl_friedrich_gauss_facts/827/

https://rabungapalgebraiii.wikispaces.com/Arithmetic+Sequences+and+Series

http://www.math.wichita.edu/MathCircle/docs/egyptian_arithmetic.pdf

Stay Focused

From Kirk Cousins, quarterback of the Washington Redskins:

Sometimes our guests ask why I have this hanging above my desk. It’s an old high school math quiz when I didn’t study at all and got a C+… just a subtle reminder to me of the importance of preparation. If I don’t prepare I get C’s!

Source: https://www.facebook.com/redskins/photos/a.118304319573.96677.102381354573/10155470824244574/?type=3&theater

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 Jessica Martinez. Her topic, from Precalculus: arithmetic sequences.

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What interesting things can you say about the people who contributed to the discovery and/or the development of this topic?

Perhaps some of us realized that we were pretty good at math at a young age, though I wonder if anyone was as good at math, or as fast a learner, as Carl Friedrich Gauss was. When Gauss was two he taught himself how to read; when he was three he checked and corrected his dad’s math whenever his father was calculating the payroll; and probably one of the most famous stories about Gauss was that when he was 9 or 10 he created a formula for an arithmetic progression just by glancing at a problem, which ultimately helped Gauss to start his lifelong education and career in mathematical theory. Gauss was sitting in an arithmetic class taught by a man named Buttner, who was said to dislike teaching peasant children, but he was so surprised and impressed that Gauss correctly calculated the solution to the sequence problem that Buttner started to take Gauss under his and wing and help him with his education. Turns out that the formula created by Gauss can be used to find any arithmetic progression. Later on, Gauss eventually earned a doctorate at the age of 22 with the financial help of the Duke of Brunswick; his dissertation was about the fundamental theory of algebra. Gauss had numerous and important contributions to the field of mathematics, but I won’t state them here for the fear of feeling highly insignificant to one of the greatest mathematicians of all time.

 

 

 

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How has this topic appeared in pop culture (movies, TV, current music, video games, etc.)?

This is kind of a cool tidbit for any music lovers and any Led Zeppelin fans. So Led Zeppelin III had a crazy array of imagery on its album sleeve and volvelle, which was great way to further engage people who listened to their music. However, we know that records are a thing of past, replaced by CDs and now online streaming. So a musician named Bill Baird, who was inspired by the themes of Led Zeppelin’s third album, created a way to make his website and album just as aesthetically pleasing and interesting as LZ by using mathematical equations and formulas. The site where you can listen to his music has a sort of kaleidoscope design (inspired by LZ’s sundials and astronomical designs) that changes as the music plays, where the artwork “mimics the music” since music themes and sounds are constantly changing over time. However, even more interesting is that the music is different for every listener. The site uses an arithmetic sequence formula based on the listener’s location and the time they accessed the site along with the hand-mixed tracks by Bill to create a unique sequence (and thus a unique album) for every listener. If the database creates an already used track, it starts the process over again until it gets a new sequence.

 

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How could you as a teacher create an activity or project that involves your topic?

One of my previous teachers suggested this as a great real world problem for arithmetic sequences: Halley’s Comet. The video itself is just a short clip in order to grab my students’ attention; the video describes the comet’s path, its velocity, and its appearances in history with fancy graphics and imaging which can appeal to the science and space lovers in my class along with history lovers. The path of the comet brings it around visible to earth about every 75 years. After we covered some basics on arithmetic sequences, I can present this video to my class and have them research the comet and how it can be represented with a sequence. Some of the questions I could ask them to answer could be: when did it last come? What are the next 3, 4, 5, etc. years it will visit? Have you or will you see it in your lifetime? Calculate its 50th visit from its last visit, its 100th visit? I could then challenge them to find some other natural phenomena that also follows an arithmetic sequence.

 

References

[Video file]. (2015, January 10). In The Legacy of Halley’s Comet. Retrieved November 18, 2016, from https://youtu.be/elsRH_utRdo

Carl Friedrich Gauss. (n.d.). Retrieved November 18, 2016, from http://www.sonoma.edu/math/faculty/falbo/gauss.html

Dial, C. (2016, October 27). Album Turns Into Something New Each Time It’s Played. Retrieved November 18, 2016, from http://www.psfk.com/2016/10/music-album-bill-baird-algorithm.html

Gauss: The Prince of Mathematics | Brilliant Math & Science Wiki. (n.d.). Retrieved November 18, 2016, from https://brilliant.org/wiki/gauss-the-prince-of-mathematics/

Howell, E. (2013, February 20). Halley’s Comet: Facts About the Most Famous Comet. Retrieved November 18, 2016, from http://www.space.com/19878-halleys-comet.html

 

My Mathematical Magic Show: Part 5d

Last March, on Pi Day (March 14, 2015), I put together a mathematical magic show for the Pi Day festivities at our local library, compiling various tricks that I teach to our future secondary teachers. I was expecting an audience of junior-high and high school students but ended up with an audience of elementary school students (and their parents). Still, I thought that this might be of general interest, and so I’ll present these tricks as well as the explanations for these tricks in this series. From start to finish, this mathematical magic show took me about 50-55 minutes to complete. None of the tricks in this routine are original to me; I learned each of these tricks from somebody else.

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Though this wasn’t part of my Pi Day magic show, I recently read an interesting variant on my fourth trick. The next time I do a mathematical show, I’ll do this trick next — not to amaze and stun my audience, but to see if my audience can figure out why it works. Each member of the audience will need to have a calculator (a basic four-function calculator will suffice). Here’s the patter:

I want you to take out your calculator. Using only the digits 1 through 9, there are three rows, three columns, and two diagonals. I want you to pick either a row, a column, or a diagonal. Then I want you to enter a three digit number using those numbers. For example, if you chose the first row, you can enter 123 or 312 or 231 or any three-digit number using each digit once.

calculator

Now, I want you to multiply this number by another three-digit number. So hit the times button.

(pause)

Now, choose another row, column, or diagonal and type in another three-digit number, using each of the three digits once.

(pause)

Now hit the equals button to multiply those two numbers together.

(pause)

Is everyone done? The product you just computed should have either five or six digits. I want you to concentrate on one of those digits. Just make sure that you concentrate on a digit other than zero, because zero is boring. So concentrate on a nonzero digit.

(pause)

(I point to someone.) Without telling me the digit you chose, please tell me the other digits in your product.

The audience member will say something like, “3, 7, 9, and 2.” To which I’ll reply in three seconds or less, “The number you chose was 6.”

Then I’ll turn to someone else and ask which numbers were not scratched out. She’ll say something like, “1, 1, 9, 7, and 2.” I’ll answer, “The number you chose was 7.”

And then I’ll repeat this a few times, and everyone’s amazed that I knew the different numbers that were chosen.

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Clearly this works using the same logic as my fourth magic trick: the product is always a multiple of 9, and so I can add the digits to figure out the missing digit. The more interesting question is: Why is the product always a multiple of 9?

This works because each of the factors of the product is a multiple of 3. Let’s take another look at the calculator.

calculatorIf the first row is chosen, the sum of the digits is 1+2+3 = 6, a multiple of 3. And it doesn’t matter if the number is 123 or 312 or 231… the order of the digits is unimportant.

If the second row is chosen, the sum of the digits is 4+5+6 = 15, a multiple of 3.

If the third row is chosen, the sum of the digits is 7+8+9 = 24, a multiple of 3.

If the first column is chosen the sum of the digits is 1+4+7=12, a multiple of 3.

If the second column is chosen, the sum of the digits is 2+5+8 = 15, a multiple of 3.

If the third column is chosen, the sum of the digits is 3+6+9 = 18, a multiple of 3.

If one diagonal is chosen, the sum of the digits is 1+5+9 = 15, a multiple of 3.

If the other diagonal is chosen, the sum of the digits is 3+5+7 = 15, a multiple of 3.

This can be stated more succinctly using algebra. The digits in each row, column, and diagonal form an arithmetic sequence. For each row, the common difference is 1. For each column, the common difference is 3. And for a diagonal, the common difference is either 2 or 4. If I let a be the first term in the sequence and let d be the common difference, then the three digits are a, a + d, and a + 2d, and their sum is

a + (a+d) + (a+ 2d) = 3a + 3d = 3(a+d),

which is a multiple of 3. (Indeed, the sum is 3 times the middle number.)

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So each factor is a multiple of 3. That means the product has to be a multiple of 9. In other words, if the first factor is 3m and the second factor is 3n, where m and n are integers, their product is equal to

(3m)(3n) = 9(mn),

which is clearly a multiple of 9. Therefore, I can use the same adding-the-digits trick to identify the missing digit.

Arithmetic and Geometric Series: Index

I’m using the Twelve Days of Christmas (and perhaps a few extra days besides) to do something that I should have done a long time ago: collect past series of posts into a single, easy-to-reference post. The following posts formed my series on how I remind students about Taylor series. I often use this series in a class like Differential Equations, when Taylor series are needed but my class has simply forgotten about what a Taylor series is and why it’s important.

Part 1: Deriving the formulas for the nth term of arithmetic and geometric sequences.

Part 2: Pedagogical thoughts on conceptual barriers that students often face when encountering sequences and series.

Part 3: The story of how young Carl Frederich Gauss, at age 10, figured out how to add the integers from 1 to 100 in his head.

Part 4: Deriving the formula for an arithmetic series.

Part 5: Deriving the formula for an arithmetic series, using mathematical induction. Also, extensions to other series.

Part 6: Deriving the formula for an arithmetic series, using telescoping series. Also, extensions to other series.

Part 7: Pedagogical thoughts on assessing students’ depth of understanding the formula for an arithmetic series.

Part 8: Deriving the formula for a finite geometric series.

Part 9: Infinite geometric series and Xeno’s paradox.

Part 10: Deriving the formula for an infinite geometric series.

Part 11: Applications of infinite geometric series in future mathematics courses.

Part 12: Other commonly-arising infinite series.

 

 

 

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.

 

Formula for an arithmetic series (Part 4)

As I’ve said before, 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 sum of an arithmetic series. This topic is commonly taught in Precalculus but, in my experience, is often forgotten by students years later when needed in later classes.

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To get the idea across, consider the arithmetic series

S = 16 + 19 + 22 + 25 + 28 + 31 + 34 + 37 + 40 + 43

Now write the sum in reverse order. This doesn’t change the value of the sum, and so:

S = 43 + 40 + 37 +34+ 31 + 28 + 25 + 22 + 19 + 16

Now add these two lines vertically. Notice that 16 + 43 = 59, 19 + 40 = 59, and in fact each pair of numbers adds to 59. So

2S = 59 + 59 + 59 + 59 + 59 + 59 + 59 + 59 + 59 + 59

2S = 59 \times 10 = 590

S = 295

Naturally, this can be directly confirmed with a calculator by just adding the 10 numbers.

When I show this to my students, they often complain that there’s no way on earth that they would have thought of that for themselves. They wouldn’t have thought to set the sum equal to S, and they certainly would not have thought to reverse the terms in the sum. To comfort them, I tell them my usual tongue-in-cheek story that this idea comes from the patented Bag of Tricks. Socrates gave the Bag of Tricks to Plato, Plato gave it to Aristotle, it passed down the generations, my teacher taught the Bag of Tricks to me, and I teach it to my students.

green lineThe derivation of the general formula proceeds using the same idea. If a_1, \dots, a_n are the first n terms of an arithmetic sequence, let

S = a_1 + a_2 + \dots + a_{n-1} + a_n

Recalling the formula for an arithmetic sequence, we know that

a_2 = a_1 + d

\vdots

a_{n-1} = a_1 + (n-2)d

a_n = a_1 + (n-1)d

Substituting, we find

S = a_1 + [a_1 + d] + \dots + [a_1 + (n-2)d] + [a_1 + (n-1)d]

As above, we now return the order…

S = [a_1 + (n-1)d] + [a_1 + (n-2)d] + \dots + [a_1 + d] + a_1

… and add the two equations:

2S = [2a_1 + (n-1)d] + [2a_1 + d+(n-2)d] + \dots + [2a_1 +(n-2)d+ d] + [2a_1+(n-1)d]

2S = [2a_1 + (n-1)d] + [2a_1 + (n-1)d] + \dots + [2a_1 +(n-1)d] + [2a_1+(n-1)d]

2S = n[2a_1 + (n-1)d]

S = \displaystyle \frac{n}{2} [2a_1 + (n-1)d]

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We also note that the formula may be rewritten as

S = \displaystyle \frac{n}{2} [a_1 + \{a_1 + (n-1)d\} ]

or

S = \displaystyle \frac{n}{2} [a_1 + a_n]

This latter form isn’t too difficult to state as a sentence: the sum of a series with n  is the average of the first and last terms, multiplied by the number of terms.

Indeed, I have seen textbooks offer proofs of this formula by using the same logic that young Gauss used to find the sum 1 + 2 + \dots + 99 + 100. The “proof” goes like this: Take the terms in pairs. The first term plus the last term is a_1 + a_n. The second term plus the second-to-last term is a_2 + a_{n-1} = a_1 + d + a_n - d = a_1 + a_n. And so on. So each pair adds to a_1 + a_n. Since there are n terms, there are n/2 pairs, and so we derive the above formula for S.

You’ll notice I put “proof” in quotation marks. There’s a slight catch with the above logic: it only works if n is an even number. If n is odd, the result is still correct, but the logic to get the result is slightly different. That’s why I don’t particularly recommend using the above paragraph to prove this formula for students, even though it fits nicely with the almost unforgettable Gauss story.

That said, for talented students looking for a challenge, I would recommend showing this idea, then point out the flaw in the argument, and then ask the students to come up with an alternate proof for handling odd values of n.

Calculation of a famous arithmetic series (Part 3)

In this post, we’ll consider the calculation of a very famous arithmetic series… not because the series is particularly important, but because it’s part of a legendary story about one of the greatest mathematicians who ever lived. My frank opinion is that every math teacher should know this story. While I’m not 100% certain about small details of the story — like whether young Gauss was 9 or 10 years old when the following event happened — I’m just going to go with the story as told by the website http://www.math.wichita.edu/history/men/gauss.html.

Carl Friedrich Gauss (1777-1855) is considered to be the greatest German mathematician of the nineteenth century. His discoveries and writings influenced and left a lasting mark in the areas of number theory, astronomy, geodesy, and physics, particularly the study of electromagnetism.

Gauss was born in Brunswick, Germany, on April 30, 1777, to poor, working-class parents. His father labored as a gardner and brick-layer and was regarded as an upright, honest man. However, he was a harsh parent who discouraged his young son from attending school, with expectations that he would follow one of the family trades. Luckily, Gauss’ mother and uncle, Friedrich, recognized Carl’s genius early on and knew that he must develop this gifted intelligence with education.

While in arithmetic class, at the age of ten, Gauss exhibited his skills as a math prodigy when the stern schoolmaster gave the following assignment: “Write down all the whole numbers from 1 to 100 and add up their sum.” When each student finished, he was to bring his slate forward and place it on the schoolmaster’s desk, one on top of the other. The teacher expected the beginner’s class to take a good while to finish this exercise. But in a few seconds, to his teacher’s surprise, Carl proceeded to the front of the room and placed his slate on the desk. Much later the other students handed in their slates.

At the end of the classtime, the results were examined, with most of them wrong. But when the schoolmaster looked at Carl’s slate, he was astounded to see only one number: 5050. Carl then had to explain to his teacher that he found the result because he could see that, 1+100=101, 2+99=101, 3+98=101, so that he could find 50 pairs of numbers that each add up to 101. Thus, 50 times 101 will equal 5050.

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.