Arithmetic Sequences

Learning Outcomes

By the end of this section, you will be able to:

  • Find the common difference for an arithmetic sequence.
  • Give terms of an arithmetic sequence.
  • Write the formula for an arithmetic sequence.

The sequence {25000, 21600, 18200, 1480, 11400, 8000} are the values of a truck each year.  The values of the truck are said to form an arithmetic sequence because they change by a constant amount each year. Each term increases or decreases by the same constant value called the common difference of the sequence. For this sequence, the common difference is –3,400.

A sequence, {25000, 21600, 18200, 14800, 8000}, that shows the terms differ only by -3400.

The sequence below is another example of an arithmetic sequence. In this case, the constant difference is 3. You can choose any term of the sequence, and add 3 to find the subsequent term.

A sequence {3, 6, 9, 12, 15, ...} that shows the terms only differ by 3.

A General Note: Arithmetic Sequence

An arithmetic sequence is a sequence that has the property that the difference between any two consecutive terms is a constant. This constant is called the common difference. If [latex]{a}_{1}[/latex] is the first term of an arithmetic sequence and [latex]d[/latex] is the common difference, the sequence will be:

[latex]\left\{{a}_{n}\right\}=\left\{{a}_{1},{a}_{1}+d,{a}_{1}+2d,{a}_{1}+3d,...\right\}[/latex]

Example 1: Finding Common Differences

Is each sequence arithmetic? If so, find the common difference.

  1. [latex]\left\{1,2,4,8,16,...\right\}[/latex]
  2. [latex]\left\{-3,1,5,9,13,...\right\}[/latex]

Try It

Is the given sequence arithmetic? If so, find the common difference.

[latex]\left\{18,\text{ }16,\text{ }14,\text{ }12,\text{ }10,\dots \right\}[/latex]

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Is the given sequence arithmetic? If so, find the common difference.

[latex]\left\{1,\text{ }3,\text{ }6,\text{ }10,\text{ }15,\dots \right\}[/latex]

Writing Terms of Arithmetic Sequences

Now that we can recognize an arithmetic sequence, we will find the terms if we are given the first term and the common difference. The terms can be found by beginning with the first term and adding the common difference repeatedly. In addition, any term can also be found by plugging in the values of [latex]n[/latex] and [latex]d[/latex] into formula below.

[latex]{a}_{n}={a}_{1}+\left(n - 1\right)d[/latex]

How To: Given the first term and the common difference of an arithmetic sequence, find the first several terms.

  1. Add the common difference to the first term to find the second term.
  2. Add the common difference to the second term to find the third term.
  3. Continue until all of the desired terms are identified.
  4. Write the terms separated by commas within brackets.

Example 2: Writing Terms of Arithmetic Sequences

Write the first five terms of the arithmetic sequence with [latex]{a}_{1}=17[/latex] and [latex]d=-3[/latex].

 

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List the first five terms of the arithmetic sequence with [latex]{a}_{1}=1[/latex] and [latex]d=5[/latex] .

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How To: Given the first term and any other term in an arithmetic sequence, find a given term.

  1. Substitute the values given for [latex]{a}_{1},{a}_{n},n[/latex] into the formula [latex]{a}_{n}={a}_{1}+\left(n - 1\right)d[/latex] to solve for [latex]d[/latex].
  2. Find a given term by substituting the appropriate values for [latex]{a}_{1},n[/latex], and [latex]d[/latex] into the formula [latex]{a}_{n}={a}_{1}+\left(n - 1\right)d[/latex].

Example 3: Writing Terms of Arithmetic Sequences

Given [latex]{a}_{1}=8[/latex] and [latex]{a}_{4}=14[/latex] , find [latex]{a}_{5}[/latex].

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Given [latex]{a}_{3}=7[/latex] and [latex]{a}_{5}=17[/latex] , find [latex]{a}_{2}[/latex] .

How To: Given the first several terms for an arithmetic sequence, write an explicit formula.

  1. Find the common difference, [latex]{a}_{2}-{a}_{1}[/latex].
  2. Substitute the common difference and the first term into [latex]{a}_{n}={a}_{1}+d\left(n - 1\right)[/latex].

Example 5: Writing the nth Term Explicit Formula for an Arithmetic Sequence

Write an explicit formula for the arithmetic sequence.

[latex]\left\{2\text{, }12\text{, }22\text{, }32\text{, }42\text{, }\dots \right\}[/latex]

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Write an explicit formula for the following arithmetic sequence.

[latex]\left\{50,47,44,41,\dots \right\}[/latex]

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Finding the Number of Terms in a Finite Arithmetic Sequence

Explicit formulas can be used to determine the number of terms in a finite arithmetic sequence. We need to find the common difference, and then determine how many times the common difference must be added to the first term to obtain the final term of the sequence.

How To: Given the first three terms and the last term of a finite arithmetic sequence, find the total number of terms.

  1. Find the common difference [latex]d[/latex].
  2. Substitute the common difference and the first term into [latex]{a}_{n}={a}_{1}+d\left(n - 1\right)[/latex].
  3. Substitute the last term for [latex]{a}_{n}[/latex] and solve for [latex]n[/latex].

Example 6: Finding the Number of Terms in a Finite Arithmetic Sequence

Find the number of terms in the finite arithmetic sequence.

[latex]\left\{8\text{, }1\text{, }-6\text{, }\dots\text{, }-41\right\}[/latex]

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Find the number of terms in the finite arithmetic sequence.

[latex]\left\{6,11,16,\dots,56\right\}[/latex]

Solving Application Problems with Arithmetic Sequences

In many application problems, it often makes sense to use an initial term of [latex]{a}_{0}[/latex] instead of [latex]{a}_{1}[/latex]. In these problems, we alter the explicit formula slightly to account for the difference in initial terms. We use the following formula:

[latex]{a}_{n}={a}_{0}+dn[/latex]

Example 7: Solving Application Problems with Arithmetic Sequences

A five-year old child receives an allowance of $1 each week. His parents promise him an annual increase of $2 per week.

  1. Write a formula for the child’s weekly allowance in a given year.
  2. What will the child’s allowance be when he is 16 years old?

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A woman decides to go for a 10-minute run every day this week and plans to increase the time of her daily run by 4 minutes each week. Write a formula for the time of her run after n weeks. How long will her daily run be 8 weeks from today?