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Recognizing Arithmetic Sequences

A sequence is a finite or infinite ordered list of numbers, also called terms. An example of a sequence is 1,4,7,10,13, 1,\, 4, \, 7, \, 10, \, 13, \ldots

The dots indicate that the sequence continues infinitely, following the same pattern. In this sequence, the difference between each term is 3.3. Any sequence where the difference between consecutive terms is constant, is an arithmetic sequence.
Concept

Patterns

In mathematics, a pattern describes a repeated change. In the example below, matches have been placed together to create three figures.

Monster1.svg

It is possible to find a pattern? Each figure has one more triangle than the last. Therefore, the next figure should have 44 triangles.

Monster 684685.svg

The number of matches increases by 2,2, and the pattern will continue for the next figures. The first figure has 33 matches, the second 5,5, etc.

Monster 2.svg

Using this pattern, the fourth figure should have 99 matches, the fifth 11,11, the sixth 13,13, etc. It can be described as the following arithmetic sequence:

3,5,7,9,11,13, 3,5,7,9,11,13,\ldots
Concept

Arithmetic Sequence

For an arithmetic sequence, the difference between consecutive terms is constant. Meaning, the difference between the first and second term is the same as the difference between the second and the third term, and so forth. This difference is called the common difference and is usually denoted with d.d. An example of an arithmetic sequence is the following.

Here, the common difference is d=2.d=2.
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Exercise

Is the following sequence arithmetic? If so, write the next three terms in the sequence. 45, 39, 33, 27, 21,  45,\ 39,\ 33,\ 27,\ 21,\ \ldots

Show Solution
Solution

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. 39-45 = \text{-}6. The difference is negative, meaning that the second term is 66 less than the first. If this is an arithmetic sequence the next term should be 396=33, 39-6=33, and it is! Furthermore, 2727 is 66 less than 33,33, and 2121 is 66 less than 27.27. Thus, the sequence is arithmetic. To find the next term we subtract 66 from 21:21\text{:} 216=15. 21-6=15. The next term is 15.15. The one after that is 156=9,15-6=9, and the one after that is 96=3.9-6=3. To summarize, the sequence is arithmetic, and the next three terms are 15,15, 9,9, and 3.3.

Concept

Arithmetic Sequences and Linear Functions

Consider the arithmetic sequence 1, 2.5, 4, 5.5, 7,  1,\ 2.5,\ 4,\ 5.5,\ 7, \ \ldots Here, the common difference is 1.5,1.5, and the terms can be illustrated in a table, where nn represents the term number and ana_n represents the term.

Because their terms change by a constant amount, arithmetic sequences show a linear relationship. Here, the common difference d=1.5d=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.
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Exercise

In a theater, there are 1010 rows of seats. The first row has 1111 seats, and for each subsequent row, the number of seats increases by 3.3. Write an arithmetic sequence to represent the number of seats in each row, then graph the sequence.

Show Solution
Solution

To begin, we can start with the first row. It is given that this row has 1111 seats. Since each row has 33 more seats than the previous row, we know the second row has 1414 seats. 11+3=14. 11+3=14. Similarly, the third row has 1717 seats, because 14+3=17.14+3=17. Continuing this pattern for the remaining rows, we can write the following sequence. 11, 14, 17, 20, 23, 26, 29, 32, 35, 38. 11,\ 14,\ 17,\ 20,\ 23,\ 26,\ 29,\ 32,\ 35,\ 38. To graph the sequence, we can let xx be the row number and yy 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 D={1,2,3,4,5,6,7,8,9,10}, D=\{1,2,3,4,5,6,7,8,9,10\}, and the range is R={11,14,17,20,23,26,29,32,35,38}. R=\{11,14,17,20,23,26,29,32,35,38\}.

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