Arithmetic Sequences: Learn It 1

  • Find the regular interval between terms in a simple sequence and use it to write the sequence’s terms
  • Use recursive and explicit formulas to represent and analyze arithmetic sequences

Terms of an Arithmetic Sequence

Companies often make large purchases, such as computers and vehicles, for business use. The book-value of these supplies decreases each year for tax purposes. This decrease in value is called depreciation. One method of calculating depreciation is straight-line depreciation, in which the value of the asset decreases by the same amount each year.

As an example, consider a woman who starts a small contracting business. She purchases a new truck for [latex]$25,000[/latex]. After five years she estimates that she will be able to sell the truck for [latex]$8,000[/latex].
[latex]\\[/latex]
The loss in value of the truck will therefore be [latex]$17,000[/latex], which is [latex]$3,400[/latex] per year for five years.
[latex]\\[/latex]
The truck will be worth [latex]$21,600[/latex] after the first year; [latex]$18,200[/latex] after two years; [latex]$14,800[/latex] after three years; [latex]$11,400[/latex] after four years; and [latex]$8,000[/latex] at the end of five years.

The values of the truck in the example 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 [latex]–3,400[/latex]. You can choose any term of the sequence, and subtract [latex]3,400[/latex] to find the subsequent term.

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

arithmetic sequence

An arithmetic sequence is a sequence where the difference between consecutive terms is always the same.

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

The difference between consecutive terms, [latex]d[/latex], and is called the common difference, for [latex]n[/latex] greater than or equal to two.

 

The sequence below is another example of an arithmetic sequence. In this case the constant difference is [latex]3[/latex]. You can choose any term of the sequence, and add [latex]3[/latex] to find the subsequent term.A sequence {3, 6, 9, 12, 15, ...} that shows the terms only differ by 3.
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]