Recognize arithmetic sequences

An arithmetic sequence is an ordered list in which the difference between every pair of consecutive terms is a constant, which may be positive, negative, or zero. Recognition includes identifying this common difference from lists, tables, or graphs and relating the sequence to the nth-term rule $a_n=a_1+(n-1)d$ , whose plotted terms follow a linear pattern. This understanding distinguishes additive change from multiplicative change and supports later work with arithmetic series and linear relationships; convergence and more abstract sequence theory are not included.

Detailed Explanation: Recognize arithmetic sequences

An arithmetic sequence is a list of numbers in which the difference between consecutive terms is always the same. This fixed difference is called the common difference, $d$ .

To recognize one:

Subtract each term from the term after it.
Check whether all the differences are equal.
If they are equal, the sequence is arithmetic.
Use

a_n=a_1+(n-1)d

to describe the term in position $n$ .

Example

Determine whether the sequence

18,\ 13,\ 8,\ 3,\ \ldots

is arithmetic, and write its nth-term rule.

Subtract consecutive terms:

13-18=-5

8-13=-5

3-8=-5

The differences are all equal, so the sequence is arithmetic. Its common difference is

d=-5.

The first term is $a_1=18$ . Substitute these values into the nth-term rule:

a_n=a_1+(n-1)d

a_n=18+(n-1)(-5)

So the rule is

\boxed{a_n=18-5(n-1)}.

The negative common difference means that the sequence decreases by $5$ each time. This is additive change: the same number is added or subtracted at every step. A sequence that is multiplied by the same number each time would not generally be arithmetic.

Learn by doing: Recognize arithmetic sequences

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Patterning - Rule for Increasing Arithmetic Pattern

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