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Finding Patterns

Humans are natural pattern finders.

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But most importantly, our ability to identify patterns allows us to make predictions. Many aspects of science are based on analyzing patterns to develop a hypothesis.

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Patterns can also help us to explain the things that we observe.

Lesson Goals

• Review how to represent a sequence in a table, a general term, and a graph.
• Discuss which representation is most appropriate for a particular problem or situation.
• Solve problems using sequences.

Try This!

In science, atoms bind together following patterns. For example, in the alkane family, single bonds bind carbon atoms together in a chain, and then hydrogen atoms bind to the carbon atoms.

The first three non-cyclic alkanes are:

You don't have to understand science to see the pattern. The first alkane, Methane, has \(1\) carbon atom and \(4\) hydrogen atoms. Then, in each subsequent term of the sequence, \(1\) carbon and \(2\) hydrogen atoms are added.

How many hydrogen atoms does a non-cyclic alkane with \(100\) carbon atoms have?

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

Review of Representing Sequences

Review of Representing Sequences

As an accumulation of our previous work with sequences, recall that we have learned how to represent a sequence in multiple ways.

Consider the following sequence

\(3,~7,~11,~15, \ldots\)

The first thing that we learned to do was to state the pattern rule of this sequence.

The pattern rule is: Start at \(3\) and add \(4\) each time

From there, we learned how to represent the sequence in a table, in a graph, and using the general term.

I want you to take a close look at the second column of the table with Term Values \(3\), \(7\), \(11\), and \(15\). Would you be able to state the pattern rule by looking only at this column?

In fact, you can, because representing the sequence in a table does not disguise the pattern rule. From this point forward, we will not necessarily be writing down the pattern rule, but instead, we're going to look for it in the table when we need it.

Check Your Understanding 1

Question (Version 1)

Which of the following represent the sequence \(5,~7,~9, \ldots\)? Select all that apply.

Answer (Version 1)

3. General Term 
   \(2n+3\)

Feedback (Version 1)

First we find the pattern rule of the sequence. 

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

Table	Graph	General Term
The table is incorrect because the term values do not all follow the pattern rule. According to the pattern rule, the \(4^{th}\) term of the sequence should be \(9+2=11\), but in the table it is \(10\).	The graph is incorrect because the ordered paris do not match the term number and term value pairs in the sequence. For example, the graph shows the ordered pair \((1,6)\), but the first term of the sequence is \(5\), not \(6\). The coordinates of this point should be \((1,5)\).	The general term is correct because if you substitute the term numbers \(1\), \(2\), and \(3\), into \(2n+3\), you get \(5\), \(7\), and \(9\).

Question (Version 2)

Which of the following represent the sequence \(3,~8,~13, \ldots\)? Select all that apply.

Answer (Version 2)

Feedback (Version 2)

First we find the pattern rule of the sequence. 

Pattern rule: Start at \(3\) and add \(5\) each time

Table	Graph	General Term
The table is correct because the term values all follow the pattern rule.	The graph is correct because the \(y\)-coordinates of the points all follow the pattern rule.	The general term is incorrect because if you substitute the term numbers \(1\), \(2\), and \(3\) into \(n+5\), you get \(3\), \(1\), and \(-1\). This does not match the sequence given.

Example 1

Solution

Our first step should always be to represent the sequence in a table.

Table

In our first column of the table, we put the term numbers. In the second column, we put the term values.

Once we have a table, there are always two things that we can do. We can either graph the sequence, or we can find a general term.

I want you to take some time now to practice representing sequences using tables, graphs, and general terms.

In the next part of the lesson, we're going to do some more practice. But we're going to focus on how to choose the best representation when solving the problem.

Check Your Understanding 2

Question

Consider the sequence \(11,~14,~17,~20,\ldots\).

Use a table to find the general term of the sequence. As an exercise you can also graph the sequence. Use \(n\) as the variable in your answer.

Answer

\(3n +8\)

Feedback

How to Represent a Sequence

Example 2

Think about the different ways to represent a sequence: in a table, in a graph, or as a general term.

Does it matter which representation we choose when trying to solve a problem?

Maybe you have a favorite one that you would like to choose all the time. Or perhaps you find yourself struggling to decide which representation is best.

Let's take some time now to do some examples and discuss which representation is helpful in each circumstance and why.

Solution

Our first step when solving a sequencing problem should be to represent the sequence in a table, along with the corresponding term numbers.

Table

Term Number	Term Value
\(1\)	\(3\)
\(2\)	\(5\)
\(3\)	\(7\)
\(4\)	\(9\)

From the table, we have three options. We can either use the table itself or choose to graph the sequence or find its general term.

In this case, neither a graph nor the general term are actually needed. Since we're looking for the \(6^{th}\) term, it's actually more efficient to apply the pattern rule and complete the next two rows of the table.

Looking at the second column in our table, with term values \(3\), \(5\), \(7\), and \(9\), we can see that the pattern rule is start at \(3\) and add \(2\) each time. Using the pattern rule, the \(5^{th}\) term is \(9\) plus \(2\), which is equal to \(11\). The \(6^{th}\) term is \(11\) plus \(2\), which is equal to \(13\).

Putting this in our table, we get

Term Number	Term Value
\(1\)	\(3\)
\(2\)	\(5\)
\(3\)	\(7\)
\(4\)	\(9\)
\(5\)	\(11\)
\(6\)	\(13\)

Therefore, the \(6^{th}\) term in the sequence is \(13\).

So here's a tip on how to know when to choose to use a table.

Use a table and the pattern rule when you have to extend the sequence by only a few terms.

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Example 3

Solution

As before, our first step when solving sequencing problem is to represent the sequence in a table along with the corresponding term numbers.

Table

Term Number	Term Value
\(1\)	\(4\)
\(2\)	\(11\)
\(3\)	\(18\)
\(4\)	\(25\)

So our first option would be to work directly with the table and apply the pattern rule until we reach a value of \(60\) in the Term Value column. 

Problem: We don't know how many additional rows are needed to find the term value \(60\). 

So we need to look for an alternative to using the table.

From a table, there's always two things we can do. We can graph the sequence or we can find a general term.

Graph

Therefore, the \(9^{th}\) term has a value of \(60\).

So how do we know when to use a graph? Well, my tip here is

Use a graph when given a term value and asked to find the term number.

Source: Light Bulb - Icons8

General Term

Before moving on to our next example, let's talk a little bit about the general term.

In this example, the graph was sufficient and the general term was not needed. However, you might be wondering why I actually chose to graph the sequence instead of finding the general term.

I can tell you that the general term of this sequence is

\(7n-3\)

It represents the term value of term \(n\).

If we know the term value is \(60\), in order to determine the term number we must solve

\(7n-3 =60\)

Right now, our only method of solving this equation is through trial and error. On the other hand, the graph can solve this type of problem without trial and error. Use this reasoning to convince yourself why right now the graph might be a better choice to solve this type of problem.

Example 4

Solution

We first represent the sequence in a table and then consider our options.

Table

Term Number	Term Value
\(1\)	\( 11\)
\(2\)	\(15\)
\(3\)	\( 19\)
\(4\)	\( 23\)

We could choose to work directly with the table and apply the pattern rule until we reach the \(89^{th}\) term.

Problem: It is not efficient to add \(85\) rows to the table.

Graph

Similarly, if we were to use a graph then the scale along the horizontal axis would have to go up to \(89\).

Problem: The horizontal axis would go to \(89\), making the graph too large.

General Term

Our final option is to find the general term. So using the table, we determine that the difference between consecutive term values, \(11\), \(15\), \(19\), and \(23\), is \(4\), so we multiply each term number by \(4\). We then notice that we need to add \(7\) in each row to get the term value.

Term Number	Term Value
\(1\)	\( \class{hl2}{\overset{\times 4}\rightarrow \; \overset{+7}\rightarrow} \ \ 11\)
\(2\)	\( \class{hl2}{\overset{\times 4}\rightarrow \; \overset{+7}\rightarrow} \ \ 15\)
\(3\)	\( \class{hl2}{\overset{\times 4}\rightarrow \; \overset{+7}\rightarrow} \ \ 19\)
\(4\)	\( \class{hl2}{\overset{\times 4}\rightarrow \; \overset{+7}\rightarrow} \ \ 23\)
\(\vdots\)	\(\vdots\)
\(n\)	\(4n +7\)

Therefore, term \(n\) has a value of \(4n\) plus \(7\).

To find the value of the \(89^{th}\) term, we must substitute \(n=89\) into the general term, \(4n+7\).

\( \begin{align*} 4n+7 &= 4(89)+7 \\ & \;= 356 + 7 \\ & \;= 363 \end{align*}\)

Therefore, the value of the \(89^{th}\) term is \(363\).

So here's a tip to help us decide when to use the general term.

Use the general term when given a term number and asked to find the missing term value. The general term is especially helpful when the term number is really, really large.

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Check Your Understanding 3

Question (Version 1)

Consider the sequence \(2,~5,~8,~11, \ldots\).

Which term has a value of \(26\)?

Answer (Version 1)

The \(9^{th}\) term has a value of \(26\).

Feedback (Version 1)

You are given a term value and asked to find its term number. In this situation, we usually solve the problem using a graph as shown below; however, since \(26\) is not that much larger than \(11\), it is also efficient to use a table. 

Thus, the \(9^{th}\) term has a value of \(26\).

Question (Version 2)

Consider the sequence \(10,~12,~14,~16, \ldots\).

What is the value of the \(15^{th}\) term?

Answer (Version 2)

The value of the \(15^{th}\) term is \(38\).

Feedback (Version 2)

You are given a term number and asked to find the missing term value. Extending the table to find the \(15^{th}\) term or drawing a graph may be time consuming, so a more efficient approach is to use the general term. 

Summary: Choosing the Best Representation

Before we move on, let's take a moment and summarize what we have learned from the previous examples.

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Solving Problems Using Sequences

Example 5

Let's now try solving some problems where you need to choose the best representation.

Take a moment and try this problem on your own.

Solution

The sequence \(12,~18,~24,~30, \ldots\) represents the perimeter of each figure in the image sequence.

Our first step is to represent the sequence in a table, along with the corresponding term numbers.

Table

Term Number	Perimeter (cm)
\(1\)	\(12\)
\(2\)	\( 18\)
\(3\)	\( 24\)
\(4\)	\( 30\)

We are given the term number and asked to find the perimeter. It makes sense to choose the general term as the representation in which we use to do this. 

Source: Light Bulb - Icons8

General Term

The difference between consecutive term values, \(12\), \(18\), \(24\), and \(30\), is \(6\). So we multiply each term number by \(6\). Notice that we then need to add \(6\) each time to get the correct term value.

Term Number	Perimeter (cm)
\(1\)	\( \class{hl2}{\overset{\times 6}\rightarrow \; \overset{+6}\rightarrow} \quad \ 12\)
\(2\)	\( \class{hl2}{\overset{\times 6}\rightarrow \; \overset{+6}\rightarrow} \quad \ 18\)
\(3\)	\( \class{hl2}{\overset{\times 6}\rightarrow \; \overset{+6}\rightarrow} \quad \ 24\)
\(4\)	\( \class{hl2}{\overset{\times 6}\rightarrow \; \overset{+6}\rightarrow} \quad \ 30\)
\(\vdots\)	\(\vdots\)
\(n\)	\(6n +6\)

As a result, term \(n\) has a value of \(6n\) plus \(6\).

To find the perimeter of the \(20^{th}\) term, substitute \(n=20\) into the general term, \(6n+6\).

\(\begin{align*} 6n+6 &= 6(20) + 6 & \text{since } n=20 \\ &\;=120 + 6 \\ &\;=126 \end{align*}\)

Therefore, the \(20^{th}\) term has a perimeter of \(126\) cm.

Example 6

Take a moment and try this problem on your own.

Solution

The sequence \(7,~11,~15,\ldots\) represents the number of squares in each figure of the image sequence.

Our first step is to represent this sequence in a table, along with the corresponding term numbers.

Table

Term Number	Number of Squares
\(1\)	\(7\)
\(2\)	\(11\)
\(3\)	\(15\)

We are given the number of squares and asked to find the term number, it makes sense to choose a graph to solve this problem.

Source: Light Bulb - Icons8

Take a moment and convince yourself why using the table or a general term are not as helpful of options. 

Graph

So we take the information from the table and we plot it.  I want you to notice that some of my points, such as term \(1\), with \(7\) squares, is plotted between \(5\) and \(10\). Sometimes doing this sort of thing on paper can be hard to do accurately. So, if you can, make your graph larger. This will help to improve the accuracy of your graph. 

Once we have the points plotted, we then draw a line through the points and extend it. 

To determine which term value has a value of \(55\), we find \(55\) on the vertical axis, and using the grid lines, find the corresponding term number.

Therefore, the \(13^{th}\) term contains \(55\) squares.

Check Your Understanding 4

Question (Version 1)

Consider the following sequence of triangles. 

Which term number has \(29\) triangles?

Answer (Version 1)

The \(8^{th}\) term has \(29\) triangles.

Feedback (Version 1)

You are given a term value and asked to find its term number. In this situation, we usually solve the problem using a graph as shown below, however since \(29\) is not that large, it is also efficient to use a table. 

Thus, the \(8^{th}\) term has \(29\) terms.

Question (Version 2)

Consider the following sequence of circles. 

How many circles will the \(8^{th}\) term have?

Answer (Version 2)

The \(8^{th}\) term will have \(18\) circles.

Feedback (Version 2)

You are given a term number and asked to find the missing term value. Extending the table to find the \(8^{th}\) term or drawing a graph may be time consuming, so a more efficient approach is to use the general term.

Try This Problem Revisited

Let's now revisit the Try This problem.

Solution

We can pull two sequences from the information that we're given. The sequence \(1,~2,~3, \ldots\) represents the number of carbon atoms in each alkane, while the sequence \(4,~6,~8, \ldots\) represents the number of hydrogen atoms in each alkane.

Our first step is to represent all of this information in a table.

Table

Number of C	Number of H
\(1\)	\( 4\)
\(2\)	\( 6\)
\(3\)	\( 8\)

We are looking for the information about the alkane with \(100\) carbon atoms. Since \(100\) is such a large number, we know that both the table and the graph would be inefficient methods. So we need to find the general term.

General Term

The difference between the consecutive terms, \(4\), \(6\), and \(8\), in the second column is \(2\). So it makes sense that we start by multiplying each value in the first column by \(2\). Note that we must add \(2\) each time to complete the relationship, and as a result if \(n\) represents the number of carbon atoms, then the number of hydrogen atoms is \(2n+2\).

Number of C	Number of H
\(1\)	\( \class{hl2}{\overset{\times 2}\rightarrow \; \overset{+2}\rightarrow} \quad 4\)
\(2\)	\( \class{hl2}{\overset{\times 2}\rightarrow \; \overset{+2}\rightarrow} \quad 6\)
\(3\)	\( \class{hl2}{\overset{\times 2}\rightarrow \; \overset{+2}\rightarrow} \quad 8\)
\(\vdots\)	\(\vdots\)
\(n\)	\(2n +2\)

We are looking for the number of hydrogen atoms in the alkane when \(n\) is equal to \(100\).

Substituting \(n=100\) into the general term, \(2n+2\), we get

\(\begin{align*} 2n+2 &= 2(100)+2 & \text{since } n=100 \\ &\;= 200 + 2 \\ &\;= 202 \end{align*}\)

Therefore, the alkane with \(100\) carbon atoms will have \(202\) hydrogen atoms.

Take It With You

Consider the following two image sequences where squares, of side length \(1\) cm, are arranged to form different shapes.

Let's compare the corresponding images in the two sequences. That is we want to compare the two images that have the same term number.

For example, we can say that the two images with term number \(1\) have perimeter \(4\) centimetres in the first sequence and a perimeter of \(16\) centimetres in the second sequence.

1. Compare the perimeters of the two images with term number \(2\).
2. Find a term number for which the perimeters of the corresponding images in the two sequences is equal.
3. Use a graph to explain why there are no other term numbers where the two perimeters of the images are equal.