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Imagine that a set of numbers is arranged according to a certain rule, just like friends with unique qualities forming a line. For example, consider a set of numbers where each one is 3 more than the one before.
1,4,7,10,13,…
Another example can be the following ordered numbers where each number is the sum of the two before it. 1,1,2,3,5,…
These arrangements that follow a specific rule are called sequences. Exploring sequences is important for understanding the order and connections in numbers, which comes in handy when solving different math problems. Catch-Up and Review
Here are a few recommended readings before getting started with this lesson.
Explore
Describing a Pattern
Interact with the applet and try to identify a pattern between the figures. What would Figure 4 look like?
Is there an expression to represent the number of points in terms of the figure's number?
Discussion
Sequence
A sequence is an ordered list of objects or elements called terms. The terms are often represented using a variable labeled with indices that specify the positions of the terms in the sequence.
Depending on the number of terms, a sequence can be finite or infinite. Since it is not possible to list all the elements in an infinite sequence, it is common to place three dots after a few terms. These three dots indicate that the sequence continues indefinitely based on a specific pattern.
The values of the terms of a sequence can repeat.
Sequence With Repeating Termsa1-1,a20,a31,a40,a5-1,a60,a71,…
Sequences can have all sorts of patterns. The examples below use the same starting term but result in different sequences due to the differences in the patterns. Note that it is common to use letters like an, bn, cn, and so on to represent distinct sequences. Discussion
Pattern
A pattern describes a repeated change of numbers, shapes, colors, actions, or other elements. Patterns are based on a specific rule. The rule can also be used to find missing steps in the pattern. In the example below, matches have been placed together to create three figures.
Is it possible to find a pattern? Notice that each figure has one more triangle than the previous one. Therefore, the next figure should have 4 triangles.
There is also a pattern in the number of matches. In each step, the number increases by 2. The first figure has 3 matches, the second figure has 5 matches, the third figure has 7 matches, and so on.
The number of matches in the next couple of figures can be found using this pattern.
3,5,7,9,11,13,…
Example
Numerical Patterns in Books
Emily and Kevin heard whispers of a mythical library filled with magical books in their town. Their research led them to some books full of mysterious numbers left behind by earlier explorers.
a Emily and Kevin came across the following numbers on one page of a book.
2,9,16,23,…
b On another page there were the following numbers.
3,12,48,192,…
Help them identify the pattern and find the next three terms for each sequence.
Answer
a Pattern: Previous term plus 7
Next Three Terms: 30, 37, and 44
Next Three Terms: 30, 37, and 44
b Pattern: Previous term times 4
Next Three Terms: 768, 3072, and 12288
Next Three Terms: 768, 3072, and 12288
Hint
a How is one term related to the next one?
b How is one term related to the next one?
Solution
Pattern: Previous term plus 7
Now use this pattern to find the next three terms in the sequence.
Next Three Terms: 30,37, and 44
b Examine the given sequence.
Pattern: Previous term times 4
Now use this pattern to find the next three terms in the sequence.
Next Three Terms: 768,3072, and 12288
Could these numbers represent the location of the library? Kevin and Emily continue to search for clues.
Pop Quiz
Finding the Terms of a Sequence
Discussion
Special Sequences: Arithmetic Sequences
Sequences are ordered sets of numbers that follow identifiable patterns. Two types of sequences are arithmetic and geometric, each with their own unique characteristics. They play a key role in understanding the organized relationships between numbers.
Concept
Arithmetic Sequence
An arithmetic sequence is a sequence that has a constant difference between consecutive terms — that is, the difference between the first and the second term is the same as the difference between the second and the third term, and so on. This difference is called the common difference and is usually denoted with d. For example, consider the sequence of all even positive integers 
For this sequence, the common difference is d=2. It is important to note that the common difference can take on negative values as well. Consider the following arithmetic sequence where the values decrease.
This is an arithmetic sequence with a common difference of -3.
Discussion
Special Sequences: Geometric Sequences
Unlike the adding pattern in arithmetic sequences, geometric sequences multiply each term by a constant factor. The characteristics of geometric sequences will now be explored.
Concept
Geometric Sequence
A geometric sequence is a sequence in which the ratio r between consecutive terms is a nonzero constant. This ratio is called the common ratio. The following is an example geometric sequence with first term 3 and common ratio 2.
The behavior of geometric sequences generally depends on the values of the first term a1 and the common ratio r. The following table shows the effects of these parameters.
| a1>0 | a1<0 | |
|---|---|---|
| r>1 | Increasing 3 →×2 6 →×2 12 →×2 24 →×2 48… |
Decreasing -3 →×2 -6 →×2 -12 →×2 -24 →×2 -48… |
| r=1 | Constant
3 →×1 3 →×1 3 →×1 3 →×1 3… |
Constant
-3 →×1 -3 →×1 -3 →×1 -3 →×1 -3… |
| 0<r<1 | Decreasing 48 →×21 24 →×21 12 →×21 6 →×21 3… |
Increasing -48 →×21 -24 →×21 -12 →×21 -6 →×21 -3… |
| r<0 | Alternating
3 →×(-2) -6 →×(-2) 12 →×(-2) -24 →×(-2) 48… |
Alternating
-3 →×(-2) 6 →×(-2) -12 →×(-2) 24 →×(-2) -48… |
Example
A Reading Challenge for the Mysterious Library
After a while, Izabella joined Kevin and Emily on their adventure. To find the mysterious library, they decided to challenge themselves by reading the books as quickly as possible.
They each had their own reading goals. Emily aimed to increase her reading by 15 pages each day, Kevin planned to read double the pages of the previous day, and Izabella alternated between reading 15 and 25 pages every day.
a Emily read 15 pages on the first day. Classify the pattern of the pages Emily read as arithmetic, geometric, or neither.
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b Izabella read 15 pages on the first day. Classify the pattern of the pages Izabella read as arithmetic, geometric, or neither.
{"type":"choice","form":{"alts":["Arithmetic Sequence","Geometric Sequence","Neither"],"noSort":false},"formTextBefore":"","formTextAfter":"","answer":2}
c Kevin read 5 pages on the first day. Classify the pattern of the pages Kevin read as arithmetic, geometric, or neither.
{"type":"choice","form":{"alts":["Arithmetic Sequence","Geometric Sequence","Neither"],"noSort":false},"formTextBefore":"","formTextAfter":"","answer":1}
Hint
b How many pages did Izabella read on the first, second, third, and fourth days?
c How many pages did Kevin read on the first, second, third, and fourth days?
Solution
a Emily read 15 pages on the first day and then increased her daily reading by an additional 15 pages each day after that. With this information, the number of pages she read in the first four days can be shown in a table as follows.
Since there is a common difference between consecutive terms, the number of pages read forms an arithmetic sequence.
| Sequence | Classification |
|---|---|
| 15,30,45,60,… | Arithmetic Sequence |
b It is given that Izabella read 15 pages on the first day and then alternated between reading 15 and 25 pages every day. With this information in mind, the number of pages she read in the first four days can be found using a table.
In this case, there is no common difference or common ratio between consecutive terms, so the number of pages is neither arithmetic nor geometric.
| Sequence | Classification |
|---|---|
| 15,25,15,25,… | Neither arithmetic nor geometric |
c Finally, take a look at the sequence formed by the number of pages Kevin read. He read 5 pages the first day, then doubled the number of pages he read every day.
Since there is a common ratio between consecutive terms, the number of pages read forms a geometric sequence.
| Sequence | Classification |
|---|---|
| 5,10,20,40,… | Geometric Sequence |
Pop Quiz
Identifying Sequences
The following applet shows the first five terms of an infinite sequence. Analyze them carefully and determine whether the sequence is arithmetic, geometric, or neither.
Example
Graphing Sequences
As Emily, Kevin, and Izabella continued their search, the books revealed more clues leading them closer to the magical library. They noticed that the mysterious numbers left by previous adventurers seemed to follow distinct patterns.
Find the fifth term of each sequence, then use the first five terms to graph each sequence on a coordinate plane.
a 7,14,21,28,…
b 3,9,27,81,…
Answer
a Fifth Term: 35
Graph:
b Fifth Term: 243
Graph:
Hint
a To find the fifth term, determine if the sequence is arithmetic or geometric. A sequence an can be considered a function of its position n. Make a table by pairing each term with its position, then plot the ordered pairs (n,an) on the coordinate plane.
b Is the sequence arithmetic or a geometric? Make a table by pairing each term with its position. Then, plot the ordered pairs (n,an) on the coordinate plane.
Solution
a A sequence an can be considered a function of its position n. Recall that a term position takes whole number values. The terms of the given sequence can be organized in a table as follows.
In this sequence, the difference between consecutive terms is 7, so it is an arithmetic sequence with a common difference of 7. This means that the fifth term of the sequence is calculated by adding 7 to the fourth term.
To graph this sequence, start by drawing a coordinate plane where the horizontal axis represents the position n and the vertical axis represents the term an. Then, plot the ordered pairs (n,an) on the coordinate plane.
b Follow the same procedure as in Part A. First, make a table to show the terms of the given sequence.
In this sequence, the difference between the first and second terms is 6, but the difference between the second and third terms is 18. There is no common difference between consecutive terms. However, there is a common ratio between consecutive terms — 3 — so this is a geometric sequence. The fifth term of this sequence will be 3 times the fourth term.
To graph this sequence, start by drawing a coordinate plane where the horizontal axis represents the position n and the vertical axis represents the term an. Then, plot the ordered pairs (n,an) on the coordinate plane.
Looking over the completed graphs, Izabella said, The arithmetic sequence looks like a straight line, but the geometric sequence looks like a curve. I wonder if this will be important.
Discussion
Explicit Rule of Arithmetic Sequences
Every arithmetic sequence can be described by a linear function that is defined for the set of counting numbers. This function, referred to as the explicit rule of an arithmetic sequence, follows a specific general format.
an=a1+(n−1)d