What is the difference between a geometric sequence and a geometric series?

A geometric sequence is a list of terms: 2, 6, 18, 54, \ldots A geometric series is the sum of those terms: 2 + 6 + 18 + 54 + \cdots Use a_n = a_1 r^{n-1} to find individual terms; use the Geometric Series formula to find their sum.

Can the first term a_1 be negative?

Yes. If a_1 = -5 and r = 2, the sequence is -5, -10, -20, -40, \ldots The formula a_n = a_1 r^{n-1} works exactly the same; just substitute the negative value for a_1.

What happens if the common ratio r equals 1?

Every term equals a_1: the sequence is constant. For example, if a_1 = 7 and r = 1, then a_n = 7 \cdot 1^{n-1} = 7 for all n. This is technically geometric but trivial.

How do I find the common ratio if I know two terms?

If you know a_i and a_j with i < j, then r = \sqrt[j-i]{\frac{a_j}{a_i}}. For example, if a_2 = 6 and a_4 = 54, then r = \sqrt[4-2]{\frac{54}{6}} = \sqrt[2]{9} = 3.

Geometric Sequence

The formula $a_n = a_1 r^{n-1}$ finds any term in a geometric sequence, where each term is a constant multiple of the previous one.

a_n = a_1 r^{\,n-1}

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Type the problem. The solver will use Geometric Sequence where Geometric Sequence is the right tool, and tell you when it is not.

What each symbol means

What Geometric Sequence takes

Symbol	Meaning
$a_n$	The nth term of the sequence — the value you're solving for when you plug in a position number.
$a_1$	The first term of the sequence, where the sequence begins; without it, you cannot scale the entire pattern.
$r$	The common ratio between consecutive terms; multiply any term by $r$ to get the next one, and it must be nonzero or the sequence ends.
$n$	The position of the term you want, counting from 1; if $n=1$ , the formula gives you $a_1$ itself, and $n$ must be a positive integer.

When to use it

Use this formula when you need to find a specific term in a sequence where each term multiplies the previous one by the same constant ratio.

Level

Usually taught in: Algebra II · Appears on: SAT

Worked examples

1. Find a term in a geometric sequence with positive ratio

Problem

Find the 5th term of a geometric sequence where

a_1 = 2

and

r = 3

$a_n = a_1 r^{n-1}$
Start with the formula for the nth term of a geometric sequence.
$a_5 = 2 \cdot 3^{5-1}$
Substitute $a_1 = 2$ , $r = 3$ , and $n = 5$ into the formula.
$a_5 = 2 \cdot 3^4$
Simplify the exponent: $5-1=4$ .
$a_5 = 2 \cdot 81$
Evaluate the power: $3^4 = 81$ .
$a_5 = 162$
Multiply: $2 \cdot 81 = 162$ .

Answer: $a_5 = 162$

We substituted our known values into the formula and evaluated step by step. The ratio of $3$ means each term is three times the previous one: $2 \to 6 \to 18 \to 54 \to 162$ .

2. Find a term in a geometric sequence with negative ratio

Problem

Find the 4th term of a geometric sequence where

a_1 = 16

and

r = -\frac{1}{2}

$a_n = a_1 r^{n-1}$
Start with the geometric sequence formula.
$a_4 = 16 \cdot \left(-\frac{1}{2}\right)^{4-1}$
Substitute $a_1 = 16$ , $r = -\frac{1}{2}$ , and $n = 4$ .
$a_4 = 16 \cdot \left(-\frac{1}{2}\right)^3$
Simplify the exponent: $4-1=3$ .
$a_4 = 16 \cdot \left(-\frac{1}{8}\right)$
Evaluate the power: $\left(-\frac{1}{2}\right)^3 = -\frac{1}{8}$ (odd exponent preserves the negative sign).
$a_4 = -2$
Multiply: $16 \cdot \left(-\frac{1}{8}\right) = -\frac{16}{8} = -2$ .

Answer: $a_4 = -2$

The negative ratio causes signs to alternate: $16 \to -8 \to 4 \to -2$ . The sequence shrinks because the ratio's absolute value $\frac{1}{2}$ is less than $1$ .

3. Find monthly data purchases in a phone plan

Problem

Your phone plan charges a monthly fee and rates per gigabyte of data. You buy additional data bundles each month. In month 1, you purchase

5

GB of data. Each month thereafter, you double your data purchase from the previous month. How much data do you buy in month 6?

$a_n = a_1 r^{n-1}$
The amount of data bought each month forms a geometric sequence with this formula.
$a_1 = 5, \quad r = 2, \quad n = 6$
Identify the first term ( $5$ GB in month 1), the common ratio (doubling each month, so $r=2$ ), and which month we want ( $n=6$ ).
$a_6 = 5 \cdot 2^{6-1}$
Substitute these values into the formula.
$a_6 = 5 \cdot 2^5$
Simplify the exponent: $6-1=5$ .
$a_6 = 5 \cdot 32$
Evaluate the power: $2^5 = 32$ .
$a_6 = 160$
Multiply: $5 \cdot 32 = 160$ GB.

Answer: $a_6 = 160 \text{ GB}$

Your monthly data purchases follow the pattern $5 \to 10 \to 20 \to 40 \to 80 \to 160$ GB. By month 6, you are buying $160$ GB—a geometric sequence growing by a factor of $2$ each month.

Common mistakes

Where Geometric Sequence usually goes wrong

The mistake	Why it is wrong	The fix
Writing $a_n = a_1 r^n$ instead of $a_n = a_1 r^{n-1}$ .	Forgetting the $-1$ in the exponent shifts every term up by one power, giving you $r$ times too large a result.	Remember: the first term uses $n=1$ , and $r^{1-1} = r^0 = 1$ , so you get $a_1$ . The exponent is always $n-1$ , not $n$ .
Plugging in the ratio as if it is the position number, computing $a_1 r^{r-1}$ when you know both $r$ and $n$ .	The ratio $r$ and position $n$ are different quantities; confusing them gives nonsense like $a_n = 3 \cdot 3^{3-1}$ when you meant $a_3 = 3 \cdot 2^{3-1}$ .	Write out which variable is which before substituting: $a_1$ is the first term, $r$ is the common ratio, and $n$ is the position number you want.
Using $a_n = a_1 r^{n-1}$ when the sequence is arithmetic, not geometric.	Geometric sequences multiply by a constant ratio; arithmetic sequences add a constant difference. Using the wrong formula produces wildly incorrect answers.	Check: does each term equal the previous term multiplied by a constant? If yes, use $a_n = a_1 r^{n-1}$ . If each term is the previous plus a constant, use the Arithmetic Sequence formula $a_n = a_1 + (n-1)d$ instead.

Tips and when to use something else

Rewrite the formula in your problem before you substitute numbers: writing $a_5 = a_1 r^{5-1}$ first makes it harder to miscount the exponent.
If the ratio is a fraction (like $r = \frac{1}{2}$ ), the sequence shrinks. If $r$ is negative, the signs alternate. Both are valid geometric sequences.
To find the sum of the first $n$ terms, use the Geometric Series formula instead of adding up individual terms by hand.
If you only know two terms but not $a_1$ and $r$ , set up the ratio $\frac{a_j}{a_i} = r^{j-i}$ to find the ratio, then work backward to find $a_1$ .

Related formulas

Geometric SeriesSums the first n terms of a geometric sequence—those where each term is the previous one times a constant—using a single formula instead of adding.Arithmetic SequenceFind the nth term in a sequence with constant differences between consecutive terms, or determine how many steps it takes to reach a specific value.Recursive FormulaA recursive formula defines each term using the previous one, helping you find patterns and generate sequences step-by-step.Arithmetic SeriesFind the sum of an arithmetic sequence without adding every term—just use the first term, last term, and how many terms there are.

Frequently asked questions

What is the difference between a geometric sequence and a geometric series?: A geometric sequence is a list of terms: $2, 6, 18, 54, \ldots$ A geometric series is the sum of those terms: $2 + 6 + 18 + 54 + \cdots$ Use $a_n = a_1 r^{n-1}$ to find individual terms; use the Geometric Series formula to find their sum.
Can the first term $a_1$ be negative?: Yes. If $a_1 = -5$ and $r = 2$ , the sequence is $-5, -10, -20, -40, \ldots$ The formula $a_n = a_1 r^{n-1}$ works exactly the same; just substitute the negative value for $a_1$ .
What happens if the common ratio $r$ equals $1$ ?: Every term equals $a_1$ : the sequence is constant. For example, if $a_1 = 7$ and $r = 1$ , then $a_n = 7 \cdot 1^{n-1} = 7$ for all $n$ . This is technically geometric but trivial.
How do I find the common ratio if I know two terms?: If you know $a_i$ and $a_j$ with $i < j$ , then $r = \sqrt[j-i]{\frac{a_j}{a_i}}$ . For example, if $a_2 = 6$ and $a_4 = 54$ , then $r = \sqrt[4-2]{\frac{54}{6}} = \sqrt[2]{9} = 3$ .

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Reviewed 2026-09-18