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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 more than the one before.
Another example can be the following ordered numbers where each number is the sum of the two before it.
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 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.
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 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 triangles.

There is also a pattern in the number of matches. In each step, the number increases by The first figure has matches, the second figure has matches, the third figure has matches, and so on.

The number of matches in the next couple of figures can be found using this pattern.

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.

Candle-and-books.jpg

a Emily and Kevin came across the following numbers on one page of a book.
b On another page there were the following numbers.
Help them identify the pattern and find the next three terms for each sequence.

Answer

a Pattern: Previous term plus
Next Three Terms: and
b Pattern: Previous term times
Next Three Terms: and

Hint

a How is one term related to the next one?
b How is one term related to the next one?

Solution

a There are various methods to identify a pattern. One approach involves examining whether the difference between consecutive terms is the same. Take a look at the given sequence.
Sequence: 2, 9, 16, 23, ...
As shown, each subsequent term can be found by to the previous one.
Now use this pattern to find the next three terms in the sequence.
Sequence: 2, 9, 16, 23, ... with a common difference of 7.
The next three terms of the sequence are and
b Examine the given sequence.
Sequence (3, 12, 48, 192, ...) has no common difference
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.
sequence: 3, 12, 48, 192, ...
In this sequence, each term can be found by the previous term
Now use this pattern to find the next three terms in the sequence.
Sequence: 3, 12, 48, 192, ... with a common ratio of 4.
The next three terms of the sequence are and
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

The following applet shows the first five terms of an infinite sequence. Identify the pattern of the sequence and use it to find the next two terms.

Interactive applet showing different infinite sequences

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 For example, consider the sequence of all even positive integers
Arithmetic sequence: 2, 4, 6, 8, 10... with a common difference of 2.
For this sequence, the common difference is 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.
Arithmetic sequence: 54, 51, 48, 45, 42, ... with common difference of -3.
This is an arithmetic sequence with a common difference of

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 between consecutive terms is a nonzero constant. This ratio is called the common ratio. The following is an example geometric sequence with first term and common ratio
Geometric Sequence: 3, 6, 12, 24, 48, ... with a common ratio of 2
The behavior of geometric sequences generally depends on the values of the first term and the common ratio The following table shows the effects of these parameters.
Increasing
Decreasing
Constant

Constant

Decreasing
Increasing
Alternating

Alternating

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.

Children-reading-books.jpg

They each had their own reading goals. Emily aimed to increase her reading by pages each day, Kevin planned to read double the pages of the previous day, and Izabella alternated between reading and pages every day.

a Emily read pages on the first day. Classify the pattern of the pages Emily read as arithmetic, geometric, or neither.
b Izabella read pages on the first day. Classify the pattern of the pages Izabella read as arithmetic, geometric, or neither.
c Kevin read pages on the first day. Classify the pattern of the pages Kevin read as arithmetic, geometric, or neither.

Hint

a How many pages did Emily read on the first, second, third, and fourth days? How can the relationship between the terms of the sequence formed by these numbers be described?
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 pages on the first day and then increased her daily reading by an additional 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
Arithmetic Sequence
b It is given that Izabella read pages on the first day and then alternated between reading and 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
Neither arithmetic nor geometric
c Finally, take a look at the sequence formed by the number of pages Kevin read. He read pages the first day, then 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
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.

Interactive applet showing different infinite sequences

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.

Numbers-and-book.jpg

Find the fifth term of each sequence, then use the first five terms to graph each sequence on a coordinate plane.

a
b

Answer

a Fifth Term:

Graph:

b Fifth Term:

Graph:

Hint

a To find the fifth term, determine if the sequence is arithmetic or geometric. A sequence can be considered a function of its position Make a table by pairing each term with its position, then plot the ordered pairs 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 on the coordinate plane.

Solution

a A sequence can be considered a function of its position 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 so it is an arithmetic sequence with a common difference of This means that the fifth term of the sequence is calculated by adding to the fourth term.

To graph this sequence, start by drawing a coordinate plane where the horizontal axis represents the position and the vertical axis represents the term Then, plot the ordered pairs 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 but the difference between the second and third terms is There is no common difference between consecutive terms. However, there is a common ratio between consecutive terms — — so this is a geometric sequence. The fifth term of this sequence will be times the fourth term.

To graph this sequence, start by drawing a coordinate plane where the horizontal axis represents the position and the vertical axis represents the term Then, plot the ordered pairs 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.

Here, is the first term and is the common difference of the sequence. This function receives the position of a term, as an input and returns the value of the term in that position, as an output.

Proof

Justification Based on Induction
Every arithmetic sequence has a common difference Therefore, it is possible to obtain every term of the sequence by adding the common difference to the first term an appropriate number of times.
Interactive applet showing how to rewrite the first five terms of a sequence as an expression involving just the common difference and the first term
Tables can help in identifying the pattern and writing a general expression.
Rewrite

The coefficient of the common difference is always less than the value of the position This makes it possible to write an explicit rule like the following formula.

Proof

Proof by Using the Point-Slope Form of a Line
A sequence can be thought of as a set of coordinate pairs where the first coordinate is the position and the second coordinate is the term value
As the position increases by 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 That means an arithmetic sequence is a linear function with a slope
Linear function and first five terms of an arithmetic sequence
Therefore, the explicit rule for the sequence can be written by substituting the coordinate pair into the point-slope form of a line.
Finally, the explicit rule can be rewritten to the form given at the beginning of this proof.

Example

Unlocking the Door

In the heart of their town, the three friends finally reached the entrance of the mystical library.

New-world-and-book.jpg

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.
b Find the term of the sequence to help Kevin, Izabella, and Emily unlock the next chapter of their adventure.

Hint

a Is the sequence an arithmetic sequence? If so, identify its first term and common difference.
b Substitute for 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 of the sequence is and the first term is These values can be used to write the explicit rule for the arithmetic sequence.
Distribute to simplify the explicit rule.
b To find the term in the sequence, substitute into the rule and simplify.
The term of the sequence is 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.

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.

Applet generating arithmetic sequences

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.

Portrait of Leonardo Fibonacci
External credits: Unknown
The Fibonacci sequence starts with two initial terms, and 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.
To illustrate the sequence, Emily generated the first few terms as follows.
Fibonacci sequence: 1, 1, 2, 3, 5, 8, ...
With the help of the librarian, Kevin discovered how the Fibonacci sequence creates an impressive spiral pattern. He formed the spiral by drawing circular arcs that link the opposite corners of consecutive squares in the Fibonacci sequence.
The Fibonacci spiral
As the three adventurers explored the Fibonacci sequence, Izabella discovered similarities between numerical models and arrangements in nature ranging from snail shells to galaxies.
Snail shell, Cyclone Catarina, and Whirlpool Galaxy
This discovery highlighted the beauty that can be found in the simplicity of numbers.