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Try This

The Fibonacci Numbers

What do you notice when you examine the following list of numbers:

\(1,~2,~3,~5,~8,~13,~21,~34,\ldots\)

• These numbers are increasing.
• These numbers are also spreading out, meaning the gaps between the numbers are getting larger.
• But what's important is that these numbers are not random.

Take a moment and try this problem on your own.

Solution

We generate numbers in this list by adding the two previous numbers.

For example,

\(1,~2,\class{hl2}{\xrightarrow{1+2=3}}3,\class{hl2}{\xrightarrow{2+3=5}}5,\class{hl2}{\xrightarrow{3+5=8}} 8,\ldots\)

What's interesting about these numbers is they appear quite often in nature. Perhaps nature knows something about math

The Fibonacci Numbers In Nature

\(1,~2,~3,~5,~8,~13,~21,~34,\ldots\)

If you look closely at the flower head of a sunflower, specifically at the spirals in each direction, you may notice that the number of spirals is almost always one of the numbers from our list. The same can be said about the spirals that form on pine cones, broccoli florets, and other flowers, such as daisies. Even the leaves that surround plant stems follow similar arrangements involving these numbers. 

Sources: Sunflower - Racide/iStock/Getty Images Plus; Pinecone - EdnaM/iStock/Getty Images Plus; 
Broccoli - jessicahyde/iStock/Getty Images Plus; Daisy - DMaxx/iStock/Getty Images Plus

Perhaps you're already familiar with these numbers, which we call the Fibonacci numbers. And perhaps you've seen before how they are generated. But why do you think they occur so often in nature?

Lesson Goals

• Given a sequence, describe a pattern rule.
• Identify the position of an item in a sequence.
• Represent information about patterns in a table.

Try This!

Consider the following sequence: 

\(1,~2,~5,~10,~17, \ldots\)

1. Can you predict what the next three numbers in this sequence might be?
2. Can you write the rule to explain how the numbers in this sequence might be generated?

Think about this problem, then move on to the next part of the lesson.

Sequences

Definition of a Sequence

Let's jump right in and discuss what a sequence actually is.

Consider the familiar sequence:

\(0,~1,~2,~3,\ldots\)

I want you to recognize these as the whole numbers. So think about why do we represent the whole numbers using a sequence? 

• We cannot write down all of the whole numbers.
• We can write down the first few numbers, in order, and have the reader figure out the pattern that we're describing.

So when we write \(0,~1,~2,~3, \ldots\), we are saying that the first numbers in order are \(0\), \(1\), \(2\), \(3\), and that this pattern is continuing (\(\ldots\)).

But what is the pattern?

Well, to get to the next item in the list, you add \(1\):

\(0, \class{hl2}{\overset{+1}{\longrightarrow}} 1, \class{hl2}{\overset{+1}{\longrightarrow}} 2, \class{hl2}{\overset{+1}{\longrightarrow}} 3, \ldots\)

This tells us that the pattern rule for this sequence is the following:

Pattern Rule: Start at \(0\) and add \(1\) each time.

Following the pattern rule, we can get to the next few numbers in the sequence. To get that next number in the list, we continue following the pattern rule:

\(0, \class{hl2}{\overset{+1}{\longrightarrow}} 1, \class{hl2}{\overset{+1}{\longrightarrow}} 2, \class{hl2}{\overset{+1}{\longrightarrow}} 3, \class{hl2}{\overset{+1}{\longrightarrow}} 4, \class{hl2}{\overset{+1}{\longrightarrow}} 5, \class{hl2}{\overset{+1}{\longrightarrow}} 6, \ldots\)

Order Matters

If the pattern rule is clear, we can convey the entire sequence by only writing down the first few numbers.

But we have to be careful, because clearly,

ORDER MATTERS

Consider the following sequence:

\(0,~3,~1,~5, \ldots\)

This sequence also lists the whole numbers, but there is no obvious pattern.

As we continue through today's lesson, consider the following:

How many numbers do you need in order to make the rule clear?

Sequences of Images

Now we can also use sequences to summarize information about diagrams. 

For example, consider the following sequence of images where toothpicks are arranged to form adjoining squares:

Source: Toothpick - melliott1605/iStock/Getty Images Plus

• The sequence \(1,~2,~3,~4,~5, \ldots \) represents the number of squares in each image.
• The sequence \(4,~7,~10,~13,~16, \ldots \) represents the number of toothpicks required to make each image.
• The sequence \(4,~6,~8,~10,~12, \ldots \) represents the perimeter of the outer rectangle in each image.

Notice how sequences are a great way to communicate different pieces of information. 

Check Your Understanding 1

Question

Select the image sequences that match the following number sequence. Select all that apply. 

\(3, ~4, ~5, \ldots\)

Answer

3. 

   Created with GeoGebra. Author: University of Waterloo. CC BY-NC-SA 4.0. http://ggbm.at/wnxT34yy

Sequences: Finding a Pattern

Describing Patterns

Let's use the following example to look at different ways we can describe patterns.

Solution 

We could use a couple of different ways to solve this problem. So let's take a moment to outline some of our options.

Now, whether we use diagrams or sequences, we have shown that the next image in the sequence (the \(6^{th}\) image) requires \(19\) toothpicks.

So in order to answer these types of questions, we must be able to identify the pattern rule.

Example 1

Take a moment and try this problem on your own.

Solution

1. So first, we need to identify that pattern rule. 
   Notice the following:
   \(2, \class{hl2}{\overset{+5}{\longrightarrow}} 7,\class{hl2}{\overset{+5}{\longrightarrow}} 12,\class{hl2}{\overset{+5}{\longrightarrow}} 17, \ldots\)

   Pattern Rule:Start at \(2\) and add \(5\) each time.

2. Using the pattern rule, we can find the next three numbers in the sequence.
   \(2,~7,~12,~17, \class{hl2}{\overset{+5}{\longrightarrow}} 22,\class{hl2}{\overset{+5}{\longrightarrow}} 27, \class{hl2}{\overset{+5}{\longrightarrow}} 32, \ldots\)

   The next three numbers in the sequence are:  \(22\), \(27\), and \(32\).

Check Your Understanding 2

Question

Find the next \(3\) terms in the following sequence.

\(1,~7,~13,~19, \ldots\)

Answer

\(25,~31,~37\)

Feedback

Consider the given terms of the sequence.

\(1 \class{hl2}{\xrightarrow{+6}} 7 \class{hl2}{\xrightarrow{+6}} 13 \class{hl2}{\xrightarrow{+6}} 19, \ldots\)

The pattern rule is: Start at \(1\) and add \(6\) each time.

We can continue following the pattern rule to find the next \(3\) numbers in the sequence.

\(1 \class{hl2}{\xrightarrow{+6}} 7 \class{hl2}{\xrightarrow{+6}} 13 \class{hl2}{\xrightarrow{+6}} 19 \class{hl2}{\xrightarrow{+6}} \boxed{\phantom\square} \class{hl2}{\xrightarrow{+6}} \boxed{\phantom\square} \class{hl2}{\xrightarrow{+6}} \boxed{\phantom\square}, \ldots\)

Doing so, we get

\(1 \class{hl2}{\xrightarrow{+6}} 7 \class{hl2}{\xrightarrow{+6}} 13 \class{hl2}{\xrightarrow{+6}} 19 \class{hl2}{\xrightarrow{+6}} 25 \class{hl2}{\xrightarrow{+6}} 31 \class{hl2}{\xrightarrow{+6}} 37, \ldots\)

Thus, the next \(3\) terms in the sequence are \(25\), \(31\), and \(37\).

Try This Problem Revisited

Let's now take a moment and revisit the Try This problem.

Solution

1. It appears that: 
   \(1,\class{hl2}{\overset{+1}\longrightarrow}2,\class{hl2}{\overset{+3}\longrightarrow}5, \class{hl2}{\overset{+5}\longrightarrow}10, \class{hl2}{\overset{+7}\longrightarrow}17, \ldots\)

   If this pattern continues, add the next odd numberm which would be \(9\), to \(17\) to obtain the next number in the sequence. 
   \(1,\class{hl2}{\overset{+1}\longrightarrow}2,\class{hl2}{\overset{+3}\longrightarrow}5, \class{hl2}{\overset{+5}\longrightarrow}10, \class{hl2}{\overset{+7}\longrightarrow}17, \class{hl2}{\overset{+9}\longrightarrow}26, \ldots\)

   The next two numbers in the sequence are obtained by adding the next two odd numbers.

   \(1,\class{hl2}{\overset{+1}\longrightarrow}2,\class{hl2}{\overset{+3}\longrightarrow}5, \class{hl2}{\overset{+5}\longrightarrow}10, \class{hl2}{\overset{+7}\longrightarrow}17, \class{hl2}{\overset{+9}\longrightarrow}26, \class{hl2}{\overset{+11}\longrightarrow}37, \class{hl2}{\overset{+13}\longrightarrow}50, \ldots\)
2. Notice that we're adding something to each number in the sequence to get the next number. Even though we don't add the same number every time, the numbers we add still follow a clear pattern.
   Rule:

   You start at \(1\) and add the odd numbers in order, to get from one number in the sequence to the next.

The Term Number

The Term Number

We have seen a few examples of sequences involving both numbers and images. So we now want to ask ourselves, how do we describe the position of the numbers in sequences?

Solution

Well, to identify the third odd number, you would locate the number in the \(3^{rd}\) position of this sequence:  

\(1,~3,~\class{hl2}{5},~7,~9, \ldots\)

We say, "the \(3^{rd}\) term of this sequence is \(5\)."

A number's position in a sequence can help us talk about specific values in any sequence that we're working with.

If we look at our sequence of odd numbers again, we can list the term numbers:

\(\underbrace{1}_{\substack{1^{st} \\ \text{term}}}, \underbrace{3}_{\substack{2^{nd} \\ \text{term}}}, \underbrace{5}_{\substack{3^{rd} \\ \text{term}}}, \underbrace{7}_{\substack{4^{th} \\ \text{term}}}, \underbrace{9}_{\substack{5^{th} \\ \text{term}}}, \underbrace{\ldots}_{\substack{\text{more} \\ \text{terms}}}\)

Example 2

Let's now do an example using that new terminology.

Take a moment and try this problem on your own.

Solution

So recall, the pattern rule is what generates the sequence.

\(4,\class{hl2}{\overset{+4}{\longrightarrow}} 8,\class{hl2}{\overset{+4}{\longrightarrow}} 12, \class{hl2}{\overset{+4}{\longrightarrow}} 16, \ldots\)

1. Pattern Rule: Start at \(4\) and add \(4\) each time.
2. Now to find the \(6^{th}\) term in the sequence, we would need to apply the pattern rule and find the next two terms. 
   \(4,\class{hl2}{\overset{+4}{\longrightarrow}} 8,\class{hl2}{\overset{+4}{\longrightarrow}} 12, \class{hl2}{\overset{+4}{\longrightarrow}} 16, \class{hl2}{\overset{+4}{\longrightarrow}} 20 ,\class{hl2}{\overset{+4}{\longrightarrow}} 24, \ldots\)

   Therefore, the \(6^{th}\) term is \(24\).

Check Your Understanding 3

Question

Find the \(7^{th}\) term in the following sequence.

\(\)\(2,~6,~10,~14, \ldots\)

Answer

\(26\)

Feedback

Consider the given terms of the sequence.

\(2, \class{hl2}{\xrightarrow{+4}} 6, \class{hl2}{\xrightarrow{+4}} 10, \class{hl2}{\xrightarrow{+4}} 14, \ldots\)

The pattern rule is: Start at \(2\) and add \(4\) each time.

We must extend the sequence to the \(7^{th}\) term, following the pattern rule.

\(2, \class{hl2}{\xrightarrow{+4}} 6, \class{hl2}{\xrightarrow{+4}} 10, \class{hl2}{\xrightarrow{+4}} 14, \class{hl2}{\xrightarrow{+4}} \boxed{\phantom\square}, \class{hl2}{\xrightarrow{+4}} \boxed{\phantom\square}, \class{hl2}{\xrightarrow{+4}} \boxed{\phantom\square}, \ldots\)

Doing so, we get

\(2, \class{hl2}{\xrightarrow{+4}} 6, \class{hl2}{\xrightarrow{+4}} 10, \class{hl2}{\xrightarrow{+4}} 14, \class{hl2}{\xrightarrow{+4}} 18, \class{hl2}{\xrightarrow{+4}} 22, \class{hl2}{\xrightarrow{+4}} 26, \ldots\)

Thus, the \(7^{th}\) term in the sequence is \(26\).

Example 3

Once we start extending sequences and assigning term numbers to sequences, we can use a table to organize information and help reveal relationships.

As we continue looking for patterns, tables can also be used to write expressions that can describe those patterns. 

Solution

The wording of the question here can be a little bit hard to understand. But essentially, we're being asked to create a table, where the term number and the number of squares in each image are paired up.

So in the first column of our table, we're going to put the Term Number, and in the second column of the table, we're going to put the Number of Squares.

In the first row, we put the information from the first image.

If we look at that first image, the term number is \(1\), and there is \(1\) square making up the image.

In the second row, we put the information from the second image.

Here, the term number is \(2\), and the number of squares is \(4\).

In the third row, we put the information from the third image.

The term number is \(3\), and the number of squares is \(9\).

Finally, in the fourth row, we put the information from the fourth image, where the term number is \(4\), and the number of squares is \(16\).

We now have a table which pairs every term number with the number of squares in that term.

Example 4

Take a moment and try this problem on your own.

Solution 

1. We start by drawing that fourth image. Each image contains one more triangle than the image before it in the sequence. So to create the fourth image, we add one triangle to the third image, using two additional toothpicks.
   

   Source: Toothpick - melliott1605/iStock/Getty Images Plus

2. Next, we want to create the table. In the first column of our table, we're going to put the Term Number. And in the second column of the table is going to go the Number of Triangles.
   In the first row, the first term has \(3\) triangles.
   In the second row, Term \(2\) has \(4\) triangles.
   In the third row, Term \(3\) has \(5\) triangles.
   And in the fourth row, Term \(4\) has \(6\) triangles.

   Term Number	Number of Triangles
   \(1\)	\(3\)
   \(2\)	\(4\)
   \(3\)	\(5\)
   \(4\)	\(6\)

   We can now use our table to extend the sequence.
   Term \(5\) has \(1\) more triangle than Term \(4\), and so Term \(5\) is going to have \(7\) triangles.
   Term \(6\) is going to have \(1\) more triangle than Term \(5\). So Term \(6\) has \(8\) triangles. 

   Term Number	Number of Triangles
   \(1\)	\(3\)
   \(2\)	\(4\)
   \(3\)	\(5\)
   \(4\)	\(6\)
   \(5\)	\(7\)
   \(6\)	\(8\)

   Our table is now complete with the first six terms of the sequence.

Check Your Understanding 4

Question

Consider the sequence of images below.

Fill in the table to relate the term number to the number of triangles. 

Answer

Feedback

The following sequence represents the number of triangles in each term of the given image sequence. 

\(1, \class{hl2}{\xrightarrow{+4}} 5, \class{hl2}{\xrightarrow{+4}} 9, \ldots\)

The pattern rule is: Start at \(1\) and add \(4\) each time. 

We must extend the sequence to find the first \(6\) terms. 

\(1, \class{hl2}{\xrightarrow{+4}} 5, \class{hl2}{\xrightarrow{+4}} 9, \class{hl2}{\xrightarrow{+4}}, \boxed{\phantom \square} \class{hl2}{\xrightarrow{+4}},\boxed{\phantom \square} \class{hl2}{\xrightarrow{+4}} \boxed{\phantom \square},\ldots\)

Doing so, we get 

\(1, \class{hl2}{\xrightarrow{+4}} 5, \class{hl2}{\xrightarrow{+4}} 9, \class{hl2}{\xrightarrow{+4}} 13, \class{hl2}{\xrightarrow{+4}} 17, \class{hl2}{\xrightarrow{+4}} 21,\ldots\)

Take It With You

Draw a sequence of images to represent the information in the following table.

Fill in the title of the second column.