Skill: 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:

1. Subtract each term from the term after it.
2. Check whether all the differences are equal.
3. If they are equal, the sequence is arithmetic.
4. 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.