Arithmetic Sequences: Learn It 1

  • Find the common difference for an arithmetic sequence.
  • Write the formula for an arithmetic sequence.
  • Use arithmetic sequences to solve realistic scenarios

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]