Math Core

Stack paper cups and each new row adds the same number of cups. Save $15 every week and your balance grows by the same amount each time. Patterns that change by a constant amount at every step are called arithmetic sequences, and they are your first look at the linear functions of the next unit.

Sequences

A sequence is an ordered list of numbers. Each number is a term. We name the terms with a letter and a subscript that gives the position: $a_1$ is the first term, $a_2$ is the second, and $a_n$ is the $n$ th term.

For the sequence $7, 11, 15, 19, 23, \ldots$ we have $a_1 = 7$ , $a_2 = 11$ , $a_3 = 15$ , and so on. The three dots mean the pattern continues.

A sequence is really a function. The input is the position $n$ and the output is the term $a_n$ . Since positions are $1, 2, 3, \ldots$ , the domain is the positive integers. In function notation you could write $a(3) = 15$ instead of $a_3 = 15$ .

Arithmetic sequences

Definition

Arithmetic sequence

An arithmetic sequence is a sequence in which you get each term by adding the same number to the term before it. That number is the common difference, $d$ .

To find $d$ , subtract any term from the term right after it: $d = a_2 - a_1$ . In $7, 11, 15, 19, \ldots$ , the common difference is $11 - 7 = 4$ . Check that it stays the same: $15 - 11 = 4$ and $19 - 15 = 4$ .

The common difference can be negative. In $20, 14, 8, 2, \ldots$ , each term is $6$ less than the last, so $d = -6$ .

Common mistake

Always subtract in the order later term minus earlier term. For $20, 14, 8, \ldots$ , computing $20 - 14 = 6$ gives the wrong sign. The sequence goes down, so $d = 14 - 20 = -6$ .

Worked example: Is it arithmetic?

Decide whether each sequence is arithmetic. If it is, give $d$ and the next two terms.

$3, 10, 17, 24, \ldots$
$2, 6, 18, 54, \ldots$
$5, 3.5, 2, 0.5, \ldots$

Solutions.

The differences are $7, 7, 7$ . Arithmetic with $d = 7$ . Next terms: $31, 38$ .
The differences are $4, 12, 36$ , which are not equal. Not arithmetic. (Each term is multiplied by $3$ ; you'll meet sequences like this in the exponents unit.)
The differences are $-1.5, -1.5, -1.5$ . Arithmetic with $d = -1.5$ . Next terms: $-1, -2.5$ .

The recursive formula

A recursive formula tells you the first term and how to get each term from the one before it. For an arithmetic sequence:

a_1 = \text{first term}, \qquad a_n = a_{n-1} + d.

Here $a_{n-1}$ means "the term just before $a_n$ ." For $7, 11, 15, \ldots$ , the recursive formula is $a_1 = 7$ , $a_n = a_{n-1} + 4$ .

Recursive formulas describe the pattern well, but they're slow for far-off terms. To find $a_{100}$ , you'd need all $99$ terms before it.

The explicit formula

An explicit formula gives any term directly from its position. Look at how the terms of $7, 11, 15, 19, \ldots$ are built:

$n$	$a_n$	built from $a_1$
$1$	$7$	$7 + 0 \cdot 4$
$2$	$11$	$7 + 1 \cdot 4$
$3$	$15$	$7 + 2 \cdot 4$
$4$	$19$	$7 + 3 \cdot 4$

To reach the $n$ th term, you start at $a_1$ and add $d$ one time fewer than the position number: $n - 1$ times.

Explicit formula for an arithmetic sequence

a_n = a_1 + (n - 1)d

where $a_1$ is the first term and $d$ is the common difference.

For $7, 11, 15, \ldots$ : $a_n = 7 + (n - 1)(4)$ . Simplifying gives $a_n = 7 + 4n - 4 = 4n + 3$ . Now $a_{100} = 4(100) + 3 = 403$ takes one step.

Worked example: Writing and using an explicit formula

For the sequence $20, 14, 8, 2, \ldots$ , write an explicit formula and find $a_{15}$ .

Here $a_1 = 20$ and $d = -6$ .

\begin{aligned} a_n &= 20 + (n - 1)(-6) \\ &= 20 - 6n + 6 \\ &= 26 - 6n \end{aligned}

Then $a_{15} = 26 - 6(15) = 26 - 90 = -64$ .

Check with a known term: $a_4 = 26 - 24 = 2$ . Correct.

Graphing an arithmetic sequence

Plot each term as a point $(n, a_n)$ . Because the terms go up by the same amount each step, the points lie on a line. The common difference is how much the points rise for each step to the right.

The terms of 7, 11, 15, 19, 23 plotted as points (n, aₙ). They lie on the dashed line y = 4x + 3.Open in grapher →

The graph is a set of separate dots, not a solid line, because the domain is only the positive integers. There is no "term number $2.5$ ." The dashed line just shows the pattern. Notice that the explicit formula $a_n = 4n + 3$ looks exactly like the equation of that line.

Working with two known terms

If you know two terms that aren't next to each other, count the steps between them.

Worked example: Finding a sequence from two terms

In an arithmetic sequence, $a_4 = 17$ and $a_{10} = 41$ . Find the explicit formula.

Going from term $4$ to term $10$ takes $10 - 4 = 6$ steps, and the value grows by $41 - 17 = 24$ . So each step adds

d = \frac{24}{6} = 4.

To find $a_1$ , go back $3$ steps from $a_4$ : $a_1 = 17 - 3(4) = 5$ .

The formula is $a_n = 5 + (n - 1)(4) = 4n + 1$ . Check: $a_{10} = 40 + 1 = 41$ . Correct.

Worked example: Which term is it?

A theater's first row has $18$ seats, and each row has $3$ more seats than the row in front of it. Which row has $102$ seats?

The seat counts form an arithmetic sequence with $a_1 = 18$ and $d = 3$ . Set the explicit formula equal to $102$ and solve for $n$ :

\begin{aligned} 18 + (n - 1)(3) &= 102 \\ (n - 1)(3) &= 84 \\ n - 1 &= 28 \\ n &= 29 \end{aligned}

Row 29 has $102$ seats.

Tip

If solving for $n$ gives a fraction or a negative number, the value is not a term of the sequence. Positions must be positive integers.