Determine terms of an arithmetic sequence

An arithmetic sequence is a sequence in which consecutive terms differ by a constant common difference, so each term can be interpreted as the previous term increased or decreased by that fixed amount. The nth term is determined by $a_n=a_1+(n-1)d$ , allowing terms to be found from a starting term and common difference, or from known terms by identifying the constant difference; this includes positive, negative, and zero differences and does not extend to more advanced series convergence or generalized sequence theory.

Detailed Explanation: Determine terms of an arithmetic sequence

An arithmetic sequence changes by the same amount each time. This fixed amount is called the common difference, $d$ .

To find a particular term, use

a_n=a_1+(n-1)d

where:

$a_n$ is the term you want,
$a_1$ is the first term,
$n$ is the term number,
$d$ is the common difference.

Example

Find the sixth term of the sequence

18,\ 15,\ 12,\ 9,\ldots

Step 1: Identify the first term and common difference.

The first term is

a_1=18

Each term decreases by $3$ , so

d=-3

Step 2: Substitute $n=6$ into the formula.

a_6=a_1+(6-1)d

a_6=18+5(-3)

Step 3: Simplify.

a_6=18-15=3

Therefore, the sixth term is

\boxed{3}

The expression $(n-1)$ is used because the first term is already given; to reach the sixth term, the common difference is added five times.

Learn by doing: Determine terms of an arithmetic sequence

Click a topic below to practice the foundational skills you'll need, learn the steps, or master this skill

Practice with unlimited practice problems

Patterning - Term Value for Increasing Arithmetic Pattern

Earned ?