4. Sequences
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Chapter 8
4.
Sequences
| Student Learning Objectives: |
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Why
This lesson covers sequences, focusing on arithmetic and geometric types and their patterns. It explains how to identify these sequences and determine their explicit rules to find terms efficiently. The Fibonacci sequence is also introduced as a famous example of a unique mathematical pattern. These concepts help in recognizing and analyzing patterns in numbers, which is essential for problem-solving and mathematical reasoning. The lesson provides practical applications to make sequences relatable and useful in real-world scenarios.
Lesson Settings & Tools
| | 15 Theory slides |
| | 10 Exercises - Grade E - A |
| | Each lesson is meant to take 1-2 classroom sessions |
Sequences
Slide of 15
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.
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Explore
Describing a Pattern
Interact with the applet and try to identify a pattern between the figures. What would Figure 4 look like?
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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.
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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,...
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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 12 288
Next Three Terms: 768, 3072, and 12 288
Hint
a How is one term related to the next one?
b How is one term related to the next one?
Solution
As shown, each subsequent term can be found by adding 7 to the previous one. Pattern:Previous term plus 7 Now use this pattern to find the next three terms in the sequence.
The next three terms of the sequence are 30, 37, and 44. Next Three Terms:30, 37, and44
b Examine the given sequence.
In this case, there is no common difference between consecutive terms. In a case like this, the next step is to check whether the terms are multiples of each other.
In this sequence, each term can be found by multiplying the previous term by 4. Pattern:Previous term times 4 Now use this pattern to find the next three terms in the sequence.
The next three terms of the sequence are 768, 3072, and 12 288. Next Three Terms:768, 3072, and12 288 Could these numbers represent the location of the library? Kevin and Emily continue to search for clues.
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Pop Quiz
Finding the Terms of a Sequence
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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
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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.
| a_1> 0 | a_1< 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 * 12 → 24 * 12 → 12 * 12 → 6 * 12 → 3 ... |
Increasing - 48 * 12 → - 24 * 12 → - 12 * 12 → - 6 * 12 → - 3 ... |
| r < 0 | Alternating
3 * (- 2) → - 6 * (- 2) → 12 * (- 2) → - 24 * (- 2) → 48 ... |
Alternating
- 3 * (- 2) → 6 * (- 2) → - 12 * (- 2) → 24 * (- 2) → - 48 ... |
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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 |
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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.
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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 a_n 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,a_n) 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,a_n) on the coordinate plane.
Solution
a A sequence a_n 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 a_n. Then, plot the ordered pairs ( n, a_n) 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 a_n. Then, plot the ordered pairs (n,a_n) 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.
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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.
a_n = a_1 + (n-1)d
Here, a_1 is the first term and d is the common difference of the sequence. This function receives the position of a term, n, as an input and returns the value of the term in that position, a_n, as an output.
Proof
Justification Based on InductionEvery arithmetic sequence has a common difference d. Therefore, it is possible to obtain every term of the sequence by adding the common difference to the first term a_1 an appropriate number of times.
Tables can help in identifying the pattern and writing a general expression.
| n | a_n | Rewrite |
|---|---|---|
| 1 | a_1 | a_1 + 0 * d |
| 2 | a_2 | a_1 + 1 * d |
| 3 | a_3 | a_1 + 2 * d |
| 4 | a_4 | a_1 + 3 * d |
| 5 | a_5 | a_1 + 4 * d |
The coefficient of the common difference is always 1 less than the value of the position n. This makes it possible to write an explicit rule like the following formula.
a_n = a_1 + (n - 1)d
Proof
Proof by Using the Point-Slope Form of a LineA sequence can be thought of as a set of coordinate pairs where the first coordinate is the position n and the second coordinate is the term value a_n. (1,a_1), (2,a_2), (3,a_3), ...
As the position increases by 1, the value of the term increases, or decreases, by a constant. Therefore, the rate of change between two consecutive coordinate pairs is constant and equal to d. That means an arithmetic sequence is a linear function with a slope d.
Therefore, the explicit rule for the sequence can be written by substituting the coordinate pair ( 1, a_1) into the point-slope form of a line. Point-Slope Form y - y_1 = m(x - x_1) [0.8em] Explicit Rule a_n - a_1 = d(n- 1) Finally, the explicit rule can be rewritten to the form given at the beginning of this proof. a_n - a_1 = d(n-1) ⇕ a_n = a_1 + (n-1)d
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Example
Unlocking the Door
In the heart of their town, the three friends finally reached the entrance of the mystical library.
All the mysteries lay behind a door with symbols on it. To unlock the door, they had to solve one more puzzle. Help them open the door.
a Write the explicit rule for the sequence.
12,20,28,36, ...
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b Find the 8th term of the sequence to help Kevin, Izabella, and Emily unlock the next chapter of their adventure.
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Hint
a Is the sequence an arithmetic sequence? If so, identify its first term and common difference.
b Substitute 8 for n into the explicit rule.
Solution
a For an arithmetic sequence, the difference between consecutive terms is constant. Check if this is true for the given sequence.
In this sequence, the difference between consecutive terms is constant, so it is an arithmetic sequence. The common difference d of the sequence is 8 and the first term a_1 is 12. These values can be used to write the explicit rule for the arithmetic sequence. a_n= a_1+(n-1) d ⇓ a_n= 12+(n-1) 8 Distribute 8 to simplify the explicit rule.
b To find the 8th term in the sequence, substitute n=8 into the rule and simplify.
a_n=12+(n-1)8
Substitute
n= 8
a_8=12+( 8-1)8
SubTerm
Subtract term
a_8=12+(7)8
Multiply
Multiply
a_8=12+56
AddTerms
Add terms
a_8= 68
The 8th term of the sequence is 68. Kevin's hand shook as he input the final number on the lock. As the giant door swung open, the sight of the mystical library beyond took their breath away.
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Pop Quiz
Identifying a Specific Term of an Arithmetic Sequence
The following applet shows the first five terms of an infinite arithmetic sequence. Determine the explicit rule of the sequence to calculate the indicated term of the sequence.
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Closure
The Fibonacci Sequence
While exploring the magical library, Emily, Kevin, and Izabella came across a section dedicated to the Fibonacci sequence. This sequence, discovered by the Italian mathematician Leonardo Fibonacci, has occupied minds for centuries with its fascinating properties.
External credits: Unknown
The Fibonacci sequence starts with two initial terms, 1 and 1. Each subsequent term is the sum of the two preceding ones. As such, it is neither an arithmetic nor a geometric sequence. Mathematically, the sequence can be expressed as follows. F_n = F_(n-1)+F_(n-2) To illustrate the sequence, Emily generated the first few terms as follows.
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Sequences
Exercise 2.1
Heichi keeps track of the number of pages he reads in a certain amount of time. The hours spent reading and the amount of pages he reads are shown in the table.
| Hours, n | Number of Pages, a_n |
|---|---|
| 1 | 20 |
| 2 | 35 |
| 3 | 50 |
Write an explicit rule that represents the relationship between the number of pages Heichi reads and the time he spends reading in hours.
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We are given a table showing the number of hours spends reading and the corresponding number of pages Heichi reads.
| Hours, n | Number of Pages, a_n |
|---|---|
| 1 | 20 |
| 2 | 35 |
| 3 | 50 |
We want to write an explicit rule to find the number of pages Heichi reads in n hours. Let's consider just the number of pages for now. They can form an increasing sequence. We can check to see if there is a constant difference between consecutive terms.
This is an arithmetic sequence with first term 20 and common difference 15. Let's use the general formula for the explicit rule of an arithmetic sequence. a_n= a_1+(n-1) d In this rule, a_1 is the first term in a given sequence, d is the common difference from one term to the next, and a_n is the nth term in the sequence. Let's substitute a_1= 20 and d= 15 into the rule and simplify.
This rule gives us the number of pages Heichi can read in n hours.
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Solution
Ignacio records the distance he runs during various time intervals. The times spent running and the corresponding distance in meters are shown in the table.
| Time in minutes, n | Distance in meters, a_n |
|---|---|
| 3 | 400 |
| 5 | 640 |
| 7 | 880 |
Write a function that expresses the relationship between the time Ignacio spends running and the distance he covers.
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We are given a table showing the times Ignacio spent running and the corresponding distance he ran. We want to write a function to express their relationship.
| Time in minutes, n | Distance in meters, a_n |
|---|---|
| 3 | 400 |
| 5 | 640 |
| 7 | 880 |
In this table, the values in the time column increase by 2 while the values in the distance row increase by 240. If we were to think of the distances Ignacio runs as a sequence, the third, fifth, and seventh terms would be 400, 640, and 880, respectively. This means that the difference between consecutive terms would be 240÷ 2 =120, and the first term would be 400-2(120)=140.
This is an arithmetic sequence with first term 160 and common difference 120. Now we can use the general formula for the explicit rule of an arithmetic sequence. a_n= a_1+(n-1) d Let's substitute a_1=160 and d=120 into the formula and simplify.
This rule provides the distance, in meters, that Ignacio ran in n minutes.
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Solution
| Position | 3 | 4 | 5 | 6 | n |
|---|---|---|---|---|---|
| Term | 1/3 | 1/9 | 1/27 | 1/81 |
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We want to describe the value of each term as a function of its position. To achieve this, we will attempt to rewrite each given term as a numeric expression that includes its position. Let's take a look at the values.
| Position | 3 | 4 | 5 | 6 | n |
|---|---|---|---|---|---|
| Term | 1/3 | 1/9 | 1/27 | 1/81 |
Notice that each term is found by multiplying the previous term by 13 and can be expressed as a power of 13.
| Position | 3 | 4 | 5 | 6 |
|---|---|---|---|---|
| Term | 1/3 | 1/9 | 1/27 | 1/81 |
| Rewrite | 1/3 = (1/3)^1 | 1/3^2= (1/3)^2 | 1/3^3= (1/3)^3 | 1/3^4= (1/3)^4 |
As we can see, the exponents increase by one for each consecutive term. Furthermore, each exponent is 2 less than the position of that term. For example, the exponent of the third term is 1, which is the result of subtracting 2 from its position 3.
| Position | 3 | 4 | 5 | 6 |
|---|---|---|---|---|
| Term | 1/3 | 1/9 | 1/27 | 1/81 |
| Rewrite Exponents | (1/3)^1 = (1/3)^(3- 2) | (1/3)^2 = (1/3)^(4- 2) | (1/3)^3 = (1/3)^(5- 2) | (1/3)^4 = (1/3)^(6- 2) |
Considering this pattern, the value of the term in position n can be described as ( 13 )^(n- 2). a_n = ( 1/3)^(n- 2) We can substitute any natural number n to get the corresponding value of term, so this is a function of the sequence.
Let's use the function we found in Part A to find the first term of the sequence. In this formula, we will replace n with 1 and then simplify.
The first term of the sequence is 3.
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Sequences
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