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Sequences and Series

Interpreting Arithmetic Sequences

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A sequence is a finite or infinite ordered list of numbers, also called terms. An example of a sequence is
The dots indicate that the sequence continues infinitely, following the same pattern. In this sequence, the difference between each term is 3. Any sequence where the difference between consecutive terms is constant, is an arithmetic sequence.

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. Consider the following example.

Arithmetic sequence with common difference of 2:2,4,6,8,10...

For this sequence, the common difference is d=2. Furthermore, consider the following example and note that the common difference can also be negative if an arithmetic sequence is decreasing.

Arithmetic sequence with common difference of -3:54,51,48,45,42,...

Example

Is the sequence arithmetic?

fullscreen
Is the following sequence arithmetic? If so, write the next three terms in the sequence.
Show Solution expand_more
To determine if the sequence is arithmetic, we have to show that the difference between consecutive terms is constant. The difference between the first and second term is
3945=-6.
The difference is negative, meaning that the second term is 6 less than the first. If this is an arithmetic sequence the next term should be
396=33,
and it is! Furthermore, 27 is 6 less than 33, and 21 is 6 less than 27. Thus, the sequence is arithmetic. To find the next term we subtract 6 from 21:
216=15.
The next term is 15. The one after that is 156=9, and the one after that is 96=3. To summarize, the sequence is arithmetic, and the next three terms are 15, 9, and 3.

Example

Graph the sequence

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In a theater, there are 10 rows of seats. The first row has 11 seats, and for each subsequent row, the number of seats increases by 3. Write an arithmetic sequence to represent the number of seats in each row, then graph the sequence.

Show Solution expand_more
To begin, we can start with the first row. It is given that this row has 11 seats. Since each row has 3 more seats than the previous row, we know the second row has 14 seats.
11+3=14.
Similarly, the third row has 17 seats, because 14+3=17. Continuing this pattern for the remaining rows, we can write the following sequence.
To graph the sequence, we can let x be the row number and y be the number of seats in the row. Arranging the sequence in a table can help us see the points that need to be graphed.

Lastly, to graph the sequence, we can plot the points from the table. Notice that the points aren't connected. This is because the row number and the number of seats in each row both have to be whole numbers.

The domain of the sequence is
and the range is

Concept

Arithmetic Sequences and Linear Functions

Consider the arithmetic sequence
Here, the common difference is 1.5, and the terms can be illustrated in a table, where n represents the term number and represents the term.

Because their terms change by a constant amount, arithmetic sequences show a linear relationship. Here, the common difference d=1.5 can be considered the slope of the line. In fact, when plotting an arithmetic sequence in a coordinate plane, it resembles the graph of a linear function.

By making this comparison, arithmetic sequences can be considered functions. They have the same characteristics as linear functions, but where linear functions are continuous, both the domain and range for arithmetic sequences are discrete.

Rule

Explicit Rule of Arithmetic Sequences

Every arithmetic sequence can be described by a linear function known as the explicit rule.

Here, a1 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 as an output.

Proof

Justification Based on Induction
Every 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 a1 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
The use of a table helps in identifying the pattern and writting a general expression.
n Rewrite
1 a1 a1+0d
2 a2 a1+1d
3 a3 a1+2d
4 a4 a1+3d
5 a5 a1+4d

The coefficient of the common difference is always 1 less than the value of the position n. That means an explicit rule such as the following formula can be written.

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 n and the second coordinate is the term value
When 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 slope d.
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 (1,a1) 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

Find and use the explicit rule of the arithmetic sequence

fullscreen
The first five terms of an arithmetic sequence are
4,7,10,13,and 16.
Find an explicit rule describing the arithmetic sequence. Then, use the rule to find the twelfth term of the sequence.
Show Solution expand_more
To find the rule, we first need to find the common difference, d, of the sequence. We can do this by subtracting one term from the next:
d=74=3.
We also know that the first term of the sequence is 4. Substituting these pieces of information into the general form of the explicit rule gives the desired rule.

To find the twelfth term of the sequence, we can now substitute n=12 into the rule.

The twelfth term is 37.

Example

Use the terms to find the rule of the sequence

fullscreen
For an arithmetic sequence,
Write an explicit rule of the sequence and give its first six terms.
Show Solution expand_more

To begin, we must determine the common difference, d. Since we do not know the values of two consecutive terms, we cannot directly find d. However, the terms a3 and a6 are 3 positions apart, so they must differ by 3d.

This gives the equation
a6a3=3d,
which we can solve for d.
a6a3=3d
015=3d
-15=3d
-5=d
d=-5
Now that we know the common difference, we have to find the first term of the sequence, a1, to write the explicit rule. Using a3 and d, we can find a1. Knowing one term, a subsequent term can be found by adding d. Similarly, a previous term is found by subtracting d.
Repeating this, we find a1.
We now know both a1 and d, so we can find the explicit rule.
Thus, the explicit rule is We have already found the terms a1, a2, a3, and a6. Finding the last two can be done either by using the explicit rule, or by adding d to a3 and then to a4. For simplicity's sake, let's add d to a3 to find a4.
Then, a5 is found using a4.
Thus, the first six terms of the sequence are
25,20,15,10,5,and 0.

Example

Model the situation using an arithmetic sequence

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Pelle is an avid collector of pellets. During his spare time, he likes to arrange his pellets in different patterns. Today, he's chosen to place them in the shape of a triangle. The top row consists of one pellet, the second of three pellets, the third of five pellets, and so on. Write a rule where is the amount of pellets in row n. Then, use the rule to find which row has 53 pellets.

Show Solution expand_more

To begin, we can make sense of the given information. For every row, the amount of pellets increase by 2. Thus, we know that d=2. It is also given that the first row consists of 1 pellet, a1=1. Using this information, we can find the rule.

Thus, the explicit rule is Next, we can find the row that contains 53 pellets. In other words, we are looking for the n that gives This is done by substituting into the rule and solving the resulting equation for n.
53=-1+2n
54=2n
27=n
n=27

Row 27 has 53 pellets.

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