Math Core

In an arithmetic sequence you add the same number to get from one term to the next. In a geometric sequence you multiply by the same number instead. Geometric sequences are the step-by-step version of exponential functions, and they describe anything that repeatedly doubles, halves or grows by a fixed percent.

Common ratio

Look at the sequence $3, 6, 12, 24, 48, \dots$ Each term is $2$ times the term before it. Dividing any term by the previous one always gives $2$ :

\frac{6}{3} = \frac{12}{6} = \frac{24}{12} = \frac{48}{24} = 2.

Definition

Geometric sequence

A geometric sequence is a sequence in which each term after the first is found by multiplying the previous term by the same nonzero number $r$ , called the common ratio.

To find $r$ , divide any term by the term before it: $r = \dfrac{a_{n}}{a_{n-1}}$ .

The ratio can be any nonzero number:

$5, 15, 45, 135, \dots$ has $r = 3$ (the terms grow).
$64, 32, 16, 8, \dots$ has $r = \tfrac{1}{2}$ (the terms shrink toward $0$ ).
$2, -6, 18, -54, \dots$ has $r = -3$ (the signs alternate).

Arithmetic or geometric?

To classify a sequence, test both patterns:

sequence	differences	ratios	type
$4, 7, 10, 13$	$3, 3, 3$	$\tfrac{7}{4}, \tfrac{10}{7}, \tfrac{13}{10}$	arithmetic
$4, 12, 36, 108$	$8, 24, 72$	$3, 3, 3$	geometric
$1, 4, 9, 16$	$3, 5, 7$	$4, \tfrac{9}{4}, \tfrac{16}{9}$	neither

A constant difference means arithmetic; a constant ratio means geometric. Some sequences are neither.

A recursive rule

A recursive rule gives the first term and tells how to get each term from the one before it. For $3, 6, 12, 24, \dots$ :

a_1 = 3, \qquad a_n = 2 \cdot a_{n-1}.

Recursive rules are natural, but slow for far-off terms. To find $a_{30}$ you'd have to compute all $29$ terms before it.

An explicit rule

Write the terms in a way that shows how many times you've multiplied by $r$ :

term	$a_1$	$a_2$	$a_3$	$a_4$	$a_n$
value	$a_1$	$a_1 r$	$a_1 r^2$	$a_1 r^3$	$a_1 r^{n-1}$

To reach the $n$ th term, you multiply by $r$ one time fewer than the term number, because the first term uses no multiplications at all.

Formulas for a geometric sequence

With first term $a_1$ and common ratio $r$ :

\text{recursive: } a_1 \text{ given}, \ \ a_n = r \cdot a_{n-1} \qquad\qquad \text{explicit: } a_n = a_1 \cdot r^{\,n-1}

Common mistake

The exponent in the explicit rule is $n - 1$ , not $n$ . For $3, 6, 12, \dots$ the formula $a_n = 3 \cdot 2^{n-1}$ gives $a_1 = 3 \cdot 2^0 = 3$ , which is correct. Using $3 \cdot 2^n$ would give $a_1 = 6$ , which is the second term.

Worked example: Finding a far-off term

Find the $8$ th term of $5, 15, 45, 135, \dots$

The first term is $a_1 = 5$ and the ratio is $r = \dfrac{15}{5} = 3$ .

a_8 = 5 \cdot 3^{8-1} = 5 \cdot 3^7 = 5 \cdot 2187 = 10{,}935.

Worked example: A negative ratio

Write an explicit rule for $2, -6, 18, -54, \dots$ and find $a_6$ .

The ratio is $r = \dfrac{-6}{2} = -3$ , so $a_n = 2(-3)^{n-1}$ . Then

a_6 = 2(-3)^5 = 2(-243) = -486.

Put parentheses around a negative ratio. Without them, $-3^5$ would mean $-(3^5)$ , which happens to give the same value here but would be wrong for even exponents.

Connection to exponential functions

The explicit rule $a_n = a_1 \cdot r^{n-1}$ looks a lot like $f(x) = a \cdot b^x$ . A geometric sequence is an exponential function whose inputs are only the counting numbers $1, 2, 3, \dots$ If you plot the terms, the points lie on an exponential curve.

The terms 3, 6, 12, 24 plotted as points (n, aₙ) lie on the exponential curve y = 1.5 · 2^x.Open in grapher →

The difference is only where the counting starts. The curve's initial value, $1.5$ , is the value that would come before the first term: $a_1 \div r = 3 \div 2$ .

Working backward

If you know two terms but not the first term, use the fact that moving forward $k$ places multiplies by $r^k$ .

Worked example: Two terms known

In a geometric sequence, $a_2 = 12$ and $a_5 = 324$ . Find $r$ and $a_1$ .

From $a_2$ to $a_5$ is $3$ steps, so $a_5 = a_2 \cdot r^3$ :

324 = 12r^3 \quad\Rightarrow\quad r^3 = 27 \quad\Rightarrow\quad r = 3.

Then go back one step from $a_2$ : $a_1 = \dfrac{12}{3} = 4$ . Check: $4, 12, 36, 108, 324$ .

Tip

Before using a formula, write out the first few terms from your rule and compare them with the given sequence. It catches off-by-one mistakes in the exponent right away.