12.8: Sequences
Introduction
The Arcade
On the way home from school on the day of the trip downtown, a bunch of students stopped off at the arcade. It was always fun to talk and get something to eat and play a video game or two. Sam and Henry began to play a favorite game of theirs with aliens.
“That has a lot of math in it,” Sasha commented as Henry had his turn.
“How do you figure?” Henry asked.
“It just does,” Sasha said convincingly. “Think about it. In this video game, an alien splits into two aliens who then split into two more aliens every 10 minutes.”
“Good point, how many aliens there would be after they split 10 times?” Henry asked.
We can start by thinking about this as a number pattern. This lesson is all about patterns and sequences. Think about the video game and you will need to solve the sequence at the end of the lesson.
What You Will Learn
In this lesson, you will learn how to complete the following skills.
- Recognize an arithmetic sequence as having a common difference between consecutive terms.
- Extend and graph arithmetic sequences.
- Recognize a geometric sequence as having a constant common ratio between consecutive terms.
- Extend and graph geometric sequences.
- Evaluate and analyze real – world situations modeled by sequences, using a graphing calculator.
Teaching Time
I. Recognize an Arithmetic Sequence as Having a Common Difference Between Consecutive Terms
Look at this example of a sequence.
You probably saw a pattern right away. If there were another set of boxes, you’d probably guess at how many there would be.
If you saw this same pattern in terms of numbers, it would look like this:
\begin{align*} 2, 4, 6, 8, 10\end{align*}
This set of numbers is called a sequence; it is a series of numbers that follow a pattern. If there was another set of boxes, you’d probably guess there would be 12, right? Just like if you added another number to the sequence, you’d write 12. You noticed that there was a difference of 2 between each two numbers, or terms, in the sequence. When we have a sequence with a fixed number between each of the terms, we call this sequence an arithmetic sequence.
Example
What is the common difference between each of the terms in the sequence?
The difference is 5 between each number.
This is an example of an arithmetic sequence. You can see that you have to be a bit of a detective to figure out the number patterns in these examples.
II. Extend and Graph Arithmetic Sequences
Finding the difference between two terms in a sequence is one way to look at sequences. We have used tables of values for several types of equations and we have used those tables of values to create graphs. Graphs are helpful because they are visual representations of the same numbers. When values rise, we can see them rise on a graph. Let’s use the same ideas, then, to graph arithmetic sequences.
Example
Graph the sequence 2, 5, 8, 11, 14, 17,...
First convert it into a table of values with independent values being the term number and the dependent values being the actual term.
Use this table to create a graph.
You can see the pattern clearly in the graph. That is one of the wonderful things about graphing arithmetic sequences.
In the graph that we created in the example, each term was expressed as a single point. This is called discrete data—only the exact points are shown. This type of data is usually involves things that are counted in whole numbers like people or boxes. Depending on what type of situation you are graphing, you may choose to connect the points with a line. The line demonstrates that there are data points between the points that we have graphed. This is called continuous data and usually involves things like temperature or length that can change fractionally.
So, we can graph sequences and classify them as either discrete or continuous data. Yet another possibility is continuing a sequence in either direction by adding terms that follow the same pattern.
III. Recognize a Geometric Sequence as Having a Constant Common Ratio Between Consecutive Terms
Arithmetic sequences are commonplace in the world of mathematics. There are other types of sequences, though, that follow other types of patterns. Look at the boxes below.
Can you see a pattern? The boxes increase each time. Using numbers, the sequence could be written 1, 4, 16, 64. You might even guess at what would come next. Is there a common difference between them? Not really. There is a difference of 3 between the first two terms, 12 between the second and third terms, and 48 between the third and fourth terms. If you guessed that 256 would follow it’s because you figured out the pattern. You noticed that to get to the next term, you have to multiply by 4 instead of by adding a certain number.
This is a geometric sequence; it’s a sequence in which the terms are found by multiplying by a fixed number called the common ratio. In the example above, the common ratio is 4.
Once you know the common ratio, then you can figure out the next step in the pattern.
Example
What is the common ratio between each of the terms in the sequence?
The ratio is 2 between each number.
You can see how knowing the common ratio helped us with our problem solving.
IV. Extend and Graph a Geometric Sequence
Consider the following sequence:
\begin{align*} 8, 24, 72, 216, \ldots\end{align*}
Doesn’t your brain want to find the next number? You’ve probably figured out that the common ratio here is 3. So the next term in the sequence would be \begin{align*}216 \cdot 3\end{align*} or 648. You would continue the same process to find the term that follows. Or, you could divide by 3 to find the previous term.
Just as we did with arithmetic sequences, it can be useful to graph geometric sequences. We’ll use the same method as before—create a table of values and then use a coordinate plane to plot the points.
Example
The amount of memory that computer chips can hold in the same amount of space doubles every year. In 1992, they could hold 1MB. Chart the next 15 years in a table of values and show the amount of memory on the same size chip in 2007.
Year | Memory (MB) |
---|---|
1992 | 1 |
1993 | 2 |
1994 | 4 |
1995 | 8 |
1996 | 16 |
1997 | 32 |
1998 | 64 |
1999 | 128 |
2000 | 256 |
2001 | 512 |
2002 | 1024 |
2003 | 2048 |
2004 | 4096 |
2005 | 8192 |
2006 | 16384 |
2007 | 32768 |
Now let’s go back and solve the problem from the introduction.
Real-Life Example Completed
The Arcade
Here is the problem from the introduction. Reread it and then solve figure out the solution.
On the way home from school on the day of the trip downtown, a bunch of students stopped off at the arcade. It was always fun to talk and get something to eat and play a video game or two. Sam and Henry began to play a favorite game of theirs with aliens.
“That has a lot of math in it,” Sasha commented as Henry had his turn.
“How do you figure?” Henry asked.
“It just does,” Sasha said convincingly. “Think about it. In this video game, an alien splits into two aliens who then split into two more aliens every 10 minutes.”
“Good point, how many aliens there would be after they split 10 times?” Henry asked.
Solution to Real – Life Example
We can write a number pattern.
2, 4, 8, 16, 32, 64, 128, 256, 512, 1024
1024 aliens after 10 splits!
This is the answer to our problem.
Vocabulary
Here are the vocabulary words that are found in this lesson.
- Sequence
- a series of numbers that follows a pattern.
- Arithmetic Sequence
- a fixed number between each of the terms in a sequence.
- Discrete data
- only the exact points are shown.
- Continuous data
- data that changes continuously.
- Geometric Sequence
- a sequence where you find terms by multiplying a fixed number by a common ratio.
Time to Practice
Directions: Write the common difference for each sequence. If there is not a pattern, indicate this in your answer.
- -9, -7, -5, -3, -1
- 5.05, 5.1, 5.15, 5.2, 5.25
- 3, 6, 10, 15, 21, 28
- 17, 14, 11, 8, 5, 2
Directions: Solve this problem by using what you know about arithmetic sequences.
An ant colony invades the caramels in a candy store. The first day they eat a \begin{align*} \frac{1}{4}\end{align*} of a caramel, the second day \begin{align*} \frac{1}{2}\end{align*} of a caramel, the third day \begin{align*} \frac{3}{4}\end{align*}.
- What is the difference between each day?
- How many do you think they’ll eat on the fourth, fifth, and sixth days?
Directions: Find the common ratio between each term.
- -4, 20, -100, 500, -2500
- 60, 15, \begin{align*} \frac{15}{4}\end{align*}, \begin{align*} \frac{15}{64}\end{align*}
- \begin{align*}\frac{1}{8}\end{align*}, \begin{align*}\frac{1}{4}\end{align*}, \begin{align*}\frac{1}{2}\end{align*}, 1, 2
- 3, 6, 8, 9
Directions: Identify the following sequences as an arithmetic sequence, a geometric sequence, or neither. For arithmetic sequences, find the common difference. For geometric sequences, find the common ratio.
- 1, 4, 7, 10, 13
- 180, 60, 20, \begin{align*}6 \frac{2}{3}\end{align*}
- 102, 94, 86, 78
- 18, 27, 35, 43, 50
- 5, -50, 500, -5000, 50000
- \begin{align*} \frac{17}{62}\end{align*}, \begin{align*} \frac{10}{31}\end{align*}, \begin{align*} \frac{23}{62}\end{align*}, \begin{align*} \frac{13}{31}\end{align*}, \begin{align*} \frac{29}{62}\end{align*}
- You have been hired to paint a house. They give you two options for payment. Option A is for you to receive $80 up front plus $40 per day. Option B is to get paid based on the number of days you work. They offer you $3 if you work only one day, $6 if you work two days, $12 if you work three days, and so on. Make two separate tables of value to show the two options for 10 days. Which option would you prefer?
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