4 3. Sequences
4.1 Learning Goals
- Identify and describe arithmetic and geometric sequences.
- Write explicit and recursive formulas for sequences.
- Use sequence formulas to find terms and partial sums.
- Find the next term in a sequence using the Method of Successive Differences.
4.2 Key Terms and Formulas
An arithmetic sequence has a constant difference \(d\).
\[ a_n = a_1 + (n-1)d \]
Recursive form:
\[ a_n = a_{n-1}+d, \quad n\ge 2 \]
A geometric sequence has a constant ratio \(r\).
\[ a_n = a_1 r^{n-1} \]
Recursive form:
\[ a_n = r\,a_{n-1}, \quad n\ge 2 \]
Method of Successive Differences:
\[ \Delta a_n = a_{n+1}-a_n \]
If first differences are constant, the sequence is arithmetic.
Arithmetic series sum (first \(n\) terms):
\[ S_n = \frac{n}{2}(a_1 + a_n) \]
Geometric series sum (first \(n\) terms, \(r \ne 1\)):
\[ S_n = a_1\frac{1-r^n}{1-r} \]
4.3 Mini-Lecture
4.3.1 Sequences
We will jump right in.
\(3, 7, 11, 15, 19, 23, \ldots\)
A number sequence is a list of numbers having a first number, a second number, a third number, and so on. These numbers are called terms. We use the three dots (ellipsis) in the sequence to indicate it goes on in this pattern forever.
Can you figure out the next term in the sequence above?
What comes after 23:
\(3, 7, 11, 15, 19, 23, \ldots\)
- There is a pattern going on in this sequence. Some mathematical operation is happening over and over again from one term to the next.
- Each number is obtained by adding 4 to the previous term.
- \(23 + 4 = 27\), so 27 is the answer.
I want you to now do the following examples.
Determine the probable next number in these sequences:
- Example 1: \(1, 2, 4, 8, 16, \ldots\)
- Example 2: \(1, 1, 2, 3, 5, 8, \ldots\)
- Example 3: \(3, 9, 15, 21, 27, \ldots\)
- Example 4: \(5, 10, 20, 40, 80, \ldots\)
Have your answers ready?
Let us see how you did.
- Example 1: \(1, 2, 4, 8, 16, 32\)
You are multiplying by 2 to get to the next term.
- Example 2: \(1, 1, 2, 3, 5, 8, 13\)
You are adding the 2 previous terms to get to the next term.
- Example 3: \(3, 9, 15, 21, 27, 33\)
You are adding 6 to get to the next term.
- Example 4: \(5, 10, 20, 40, 80, 160\)
You are multiplying by 2 to get to the next term.
An Arithmetic Sequence is made by adding the same value each time. The value being added is called the common difference.
In Example 3 we see an arithmetic sequence and it has common difference 6:
\(3, 9, 15, 21, 27, 33\)
A Geometric Sequence is made by multiplying the same value each time. The value being multiplied is called the common ratio.
In Examples 1 and 4 we see geometric sequences and each has common ratio 2:
\(1, 2, 4, 8, 16, 32\) and \(5, 10, 20, 40, 80, 160\)
\(1, 1, 2, 3, 5, 8, 13\) is my favorite sequence. It is not arithmetic nor geometric. It has a special name. Do you remember what it is? We will talk about it soon.
I just want to mention that people use sequences a lot in various ways. You may not be given a sequence written in proper form in your everyday life, but that does not mean you cannot recognize patterns and rewrite data as sequences.
Let us say you are a fairly new influencer and partner with a certain brand in an advertising campaign. You predict the number of hits on a featured post will double each day. You round to the nearest hundred to collect your data.
This rounded data would look like:
| Day | 1 | 2 | 3 | 4 | 5 | … |
|---|---|---|---|---|---|---|
| Hits | 200 | 400 | 800 | 1600 | 3200 | … |
If your model holds true and continues this way, how many hits will you expect at the end of the week? We will predict day 7.
Write the hits as a sequence:
\(200, 400, 800, 1600, 3200, \ldots\)
The next term is 6400 (day 6), but we want day 7, so the next term is 12,800 hits (rounded to the nearest hundred).
Can you also classify this sequence as arithmetic, geometric, or neither?
Since we multiply by 2 each time, it is geometric.
I now want you to do examples where you determine whether each sequence is arithmetic, geometric, or neither.
Arithmetic, geometric, or neither?
- Example 1: \(2, 6, 18, 54, 162, \ldots\)
- Example 2: \(1, 3, 9, 27, 81, \ldots\)
- Example 3: \(3, 8, 13, 18, 23, \ldots\)
- Example 4: \(1, 3, 6, 10, 15, \ldots\)
- Example 5: \(2, 6, 22, 56, 114, \ldots\)
Done? Let us see how you did.
- Example 1: geometric, common ratio 3 (486 is the next term).
- Example 2: geometric, common ratio 3 (243 is the next term).
- Example 3: arithmetic, common difference 5 (28 is the next term).
- Example 4: neither, since there is no common difference nor common ratio (21 is the next term). We are going to talk about this special sequence later on.
- Example 5: neither, since there is no common difference nor common ratio (202 is the next term). Here it is not obvious what the next term is going to be.
You can watch me explain these examples in this video.
4.3.2 Successive Differences
Often, the method of successive differences may be applied to determine the next term in a sequence. This method is especially useful when sequences are more difficult, like
\(3, 19, 165, 771, 2503, 6483, 14409, \ldots\)
(We will figure this one out soon!)
The method of successive differences involves repeated subtraction until the difference is a constant value. Once you have a row of constant values, work backward by adding until the desired term of the original sequence is obtained.
This might sound confusing right now, but after an example or two, you will get the hang of it.
We will do the following examples:
Find the next term in \(4, 9, 16, 25, \ldots\)
Find the next term in \(1, 8, 27, 64, \ldots\)
Find the next term in \(14, 22, 32, 44, \ldots\)
Find the next term in \(5, 15, 37, 77, 141, \ldots\)
Find the next term in \(2, 6, 22, 56, 114, \ldots\)
Find the next term in \(4, 9, 16, 25, \ldots\)
I want to explain this one slowly so you feel confident for the first example. We will follow the step-by-step instructions.
- Let’s write the numbers without commas and spread out and put a blank for the number we are looking for.
4 9 16 25 ____
- Now we subtract the first term from the second term, the second from the third, and so on. Scrap: 9 - 4 = 5, 16 - 9 = 7, 25 - 16 = 9
4 9 16 25 ____ 5 7 9 ____
- Since our new row of differences is not constant, we subtract again. We subtract the first term from the second term, the second from the third, and so on. Scrap: 7 - 5 = 2, 9 - 7 = 2
4 9 16 25 ____
5 7 9 ____
2 2 ____
- Since our new row of differences is a constant, 2, we do not need to subtract anymore. This means we can start figuring out the blanks from the bottom up.
The last blank is in the row of constants so we fill that in with whatever the constant is. In this case the constant is 2, so that is the last blank.
4 9 16 25 ____ 5 7 9 ____ 2 2 2 ⬅️
- Let’s continue filling in the blanks. The next blank in the row above the 2 represents a number, 11, that gives us 2 as the difference. We can figure it out by addition: 2 + 9 = 11.
4 9 16 25 ____
5 7 9 11 ⬅️
2 2 2
- We are up to the last blank. The last blank represents a number, 36, that gives us 11 as the difference. We can figure it out by addition: 11 + 25 = 36.
4 9 16 25 36 ⬅️
5 7 9 11
2 2 2
So the next term is 36.
Let’s do one more example together step-by-step before moving a bit quicker.
- Find the next term in \(1, 8, 27, 64, \ldots\)
I want to explain slowly so you feel confident for the second example. We will follow the same step-by-step instructions with different numbers.
- Let’s write the numbers without commas and spread out and put a blank for the number we are looking for.
1 8 27 64 ____
- Now we subtract the first term from the second term, the second from the third, and so on. Scrap: 8 - 1 = 7, 27 - 8 = 19, 64 - 27 = 37
1 8 27 64 ____ 7 19 37 ____
- Since our new row of differences is not constant, we subtract again. We subtract the first term from the second term, the second from the third, and so on. Scrap: 19 - 7 = 12, 37 - 19 = 18
1 8 27 64 ____
7 19 37 ____
12 18 ____
- Since our new row of differences is not constant yet, we subtract again. We subtract the first term from the second term, the second from the third, and so on.
1 8 27 64 ____
7 19 37 ____
12 18 ____
6 6 ⬅️
- Since our new row of differences is a constant, 6, we can start figuring out the blanks from the bottom up. The next blank in the row above the 6 represents a number, 24, that gives us 6 as the difference. We can figure it out by addition: 6 + 18 = 24.
1 8 27 64 ____
7 19 37 61
12 18 24
6 6 6
- We are up to the last blank. The last blank represents a number, 125, that gives us 61 as the difference. We can figure it out by addition: 61 + 64 = 125.
1 8 27 64 125
7 19 37 61
12 18 24
6 6 6
So the next term is 125.
Use the method of successive differences to determine the next number in each sequence.
- \(14, 22, 32, 44, \ldots\)
14 22 32 44 58
8 10 12 14
2 2 2
So the next term is 58.
You can watch a video with narration of me doing this example (with help from my son): right here.
- \(5, 15, 37, 77, 141, \ldots\)
5 15 37 77 141 235
10 22 40 64 94
12 18 24 30
6 6 6
So the next term is 235.
- \(2, 6, 22, 56, 114, \ldots\)
2 6 22 56 114 202
4 16 34 58 88
12 18 24 30
6 6 6
This sequence is neither arithmetic nor geometric, and the next term is 202. You should now be able to see why using successive differences.
Use the method of successive differences to determine the next number in each sequence:
\(1, 4, 11, 22, 37, 56, \ldots\)
\(3, 19, 165, 771, 2503, 6483, 14409, \ldots\)
Now it is time to do two more, including the one from the beginning of this mini-lecture section! I believe you can do them both now. Once you finish the second one I hope you feel proud of yourself!
For the first exercise, did you get 79? If so, great. If not, check:
- First differences: \(3, 7, 11, 15, 19, \underline{\hspace{1cm}}\)
- Next row has constant 4.
- Working upward gives 23, then \(56 + 23 = 79\).
For 2., did you get 28,675? Keep trying if not. If not, check:
The constant is 120.
The first blank above 120 is 696.
First differences: \(16, 146, 606, 1732, 3980, 7926, \underline{\hspace{1cm}}\)
Second differences: \(130, 460, 1126, 2248, 3946, \underline{\hspace{1cm}}\)
Third differences: \(330, 666, 1122, 1698, \underline{\hspace{1cm}}\)
Fourth differences: \(336, 456, 576, \underline{\hspace{1cm}}\)
Fifth differences: \(120, 120, \underline{\hspace{1cm}}\)
Working upward gives 696, 2394, 6340, 14266, then \(14409 + 14266 = 28675\).
I have a video of each step on how to do an example like number 8 above (from a decade ago): HERE
4.3.3 Fibonacci Part 1
It is now time to talk about one of my favorite topics, the Fibonacci Sequence. (We will do more Fibonacci in the future!)
Fibonacci Sequence video my kids love.
Leonardo of Pisa (aka Fibonacci) wrote about a problem involving rabbits back in 1202. This rabbit problem is one of the most famous problems in elementary mathematics.
Here it is: A man put a pair of rabbits in a cage. During the first month the rabbits produced no offspring, but each month thereafter produced one new pair of rabbits. If each new pair thus produced reproduces in the same manner, how many pairs of rabbits will there be at the end of one year?
Even though the above is the famous problem, I like a different one that actually happens in nature. At the very end of this mini-lecture we will have the answer for this rabbit problem.
Let us look at a pattern that happens in nature. Male honeybees hatch from eggs which have not been fertilized, so a male bee has only one parent (a female). Female honeybees hatch from fertilized eggs, which means they have both a male and female parent.
Create a chart showing the ancestors of a random male honeybee. Keep going for about 6 generations and stop.
Notice that we start with one male honeybee, who comes from one honeybee. His mother comes from two honeybees. The next generation has three honeybees, followed by 5 honeybees, 8 honeybees, 13 honeybees, and so on. You can watch me explain the chart in this video.
You should notice you are generating the Fibonacci Sequence.
The Fibonacci Sequence is
\[ 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, \ldots \]
Can you figure out what three terms come next?
After trying, the next three are:
\[ 233, 377, 610, \ldots \]
How did we get that answer?
Each term in the Fibonacci sequence is the sum of the previous two terms. After the first two terms (both 1), each term is obtained by adding the two previous terms.
If \(F_n\) represents the Fibonacci number in the \(n\)th position in the sequence, then the recursive formula is:
\[ F_n = F_{n-1} + F_{n-2}, \quad n \ge 3. \]
Example: for the 4th Fibonacci number,
\[ F_4 = F_3 + F_2 = 2 + 1 = 3. \]
Now try these:
- Find \(F_{11}\).
- If \(F_{17}=1597\) and \(F_{18}=2584\), what is \(F_{16}\)?
- What is \(F_{19}\)? (Hint: use number 2.)
Answers:
- \(F_{11}=89\)
- \(F_{16}=987\) because \(F_{18}=F_{17}+F_{16}\) gives \(2584=1597+F_{16}\).
- \(F_{19}=4181\) because \(F_{19}=F_{18}+F_{17}=2584+1597\).
Confused with any of these three problems? Please feel free to watch me do them and explain in this video.
I have a brain teaser for you. Let us say we have a row of \(n\) chairs. People arrive in pairs and want to sit next to their partner. They can also decide not to sit and stand.
For a row of 4 chairs, there are 5 ways to fill the row (with adjacent pairs or empty chairs).
Try these:
- How many ways are there for 5 chairs?
- What about 2 chairs?
- What about 3 chairs?
- How many ways can a row of \(n\) chairs be filled in general?
Organizing results:
- 2 chairs -> 2 ways
- 3 chairs -> 3 ways
- 4 chairs -> 5 ways
- 5 chairs -> 8 ways
These are Fibonacci numbers again.
If there are \(n\) chairs, the number of ways is:
\[ F_{n+1}. \]
For example, 7 chairs gives \(F_8=21\) ways. Want to see this entire example explained? Please feel free to watch me go over the whole thing in this video.
If you want more Fibonacci (yes, I do), here are some fun extras that connect to nature, art, design, and music:
4.4 Practice
- Determine whether each sequence is arithmetic, geometric, or neither: \(2,6,18,54,\dots\) and \(7,12,17,22,\dots\).
- Write explicit and recursive formulas for \(4,9,14,19,\dots\).
- Find \(a_{12}\) for a geometric sequence with \(a_1=3\) and \(r=2\).
- Find the sum of the first 15 terms of \(10,13,16,\dots\).
- A sequence is defined by \(a_1=2\) and \(a_n=a_{n-1}+5\) for \(n\ge2\). Write the explicit formula and compute \(a_{10}\).
- A geometric sequence has \(a_1=81\) and \(r=\frac{1}{3}\). Find \(n\) such that \(a_n=1\).
- Use the Method of Successive Differences to find the next term of \(3,8,15,24,\dots\).
- Find the sum of the first 8 terms of the geometric sequence \(5,10,20,40,\dots\).
4.5 Art and Design Connections
- Believe it or not, finding the next term in a sequence is a popular kind of brain teaser.
- A game called Wordle is super popular right now (well, it was at the beginning of 2022; now many of my friends stopped but I keep going in 2026). One of my friends posted that there is a sequence version: Times Table Sequence version for you to play here.
- Sequences can be visual. We will see that soon. I made a video showing how \(1, 3, 6, 10, 15, \ldots\) can look animated.
You can also turn them into sounds and songs! Raise the volume on your device then click here to hear the melody. Thought this was super cool and a new find for me (from December 2021). I used the same sequence but feel free to play around with all different types to create different songs! (You are welcome to send me your creations.)
You also see number sequences every day in NYC, such as house and apartment numbers. We also see certain sequences in nature and food, and we will talk a lot about that in the next mini-lecture.
If I were a professor at Hogwarts, then to protect the stone I would use a difficult successive difference. You all would be able to help save the day.
Generate a stripe pattern where widths follow an arithmetic sequence and colors follow a geometric repetition schedule.
Design staircase typography with letter heights modeled by a sequence and explain the recursive rule used.
Build a photo-collage grid whose tile counts follow Fibonacci numbers, then discuss balance and focal points.
4.6 Creative Assignment
4.6.1 Creative Assignment for this Chapter
(Creative Homework Assignment #2: Number/s)
Your second creative assignment is to create an original piece of art using a number or numbers.
- You can use a single number as many times as you want or
- you can choose many different numbers or
- You can use a single number once.
You can decide which of these three options. I need to be able to see at least one number.
4.6.2 Examples and More Information
- See the module folder on our course site for examples that would get credit and bonus for this creative homework assignment.
- For information on how these assignments work; the grading rubric; and the voting you can look in Chapter 9 of this textbook or many places on our course site!
- The more effort you put in for these assignments, the more bonus you get on exams. It helps if you write how long it took you to complete your work and how you created your assignment.
4.7 Exercises
4.7.1 Exercises for this Chapter
- Make sure you are logged into your FIT Google account or else you will not view the link below.
- Once you have your answers, submit them carefully through our course site on Brightspace by the deadline.
The above are the Textbook Exercises for my MA142 students.
4.7.2 More Exercises
These questions are for anyone! They are not required for my students.
Identify the Sequence Type. For each sequence, state whether it is arithmetic, geometric, or neither. If arithmetic, give the common difference \(d\). If geometric, give the common ratio \(r\).
- \(3, 7, 11, 15, 19, \ldots\)
- \(2, 6, 18, 54, \ldots\)
- \(1, 4, 9, 16, 25, \ldots\)
- \(100, 50, 25, 12.5, \ldots\)
Write a Formula. The sequence \(5, 8, 11, 14, \ldots\) is arithmetic.
- Write an explicit formula for the \(n\)-th term.
- Write a recursive formula for the sequence.
- Use your explicit formula to find the 20th term.
Method of Successive Differences. Use the Method of Successive Differences to find the next two terms of the sequence: \(2, 5, 10, 17, 26, \ldots\)
Partial Sum. Find the sum of the first 10 terms of the geometric sequence \(1, 2, 4, 8, \ldots\)
Pop Culture Connection. Taylor Swift’s Eras Tour had a setlist that grew with each leg of the tour. Suppose the number of songs performed each night followed an arithmetic sequence: the first night she performed 22 songs, and each subsequent night she added 2 more songs to the setlist.
- Write an explicit formula for the number of songs on night \(n\).
- How many songs would she perform on night 15?
- If the total number of songs across all nights is 500, how many nights did the tour run? (Hint: use the partial sum formula and solve for \(n\).)