Arithmetic and geometric sequences worksheet for Precalculus

Precalculus sequences use large term numbers, negative ratios and formulas built from two terms that are not next to each other. Expect explicit formulas with a parenthesized ratio, which sets up the series and sigma notation that follow.

Also taught in Algebra 2 and Algebra 1, each at its own level and with its own examples.

Not quite the one you need? Same words, different skill:

Arithmetic Sequences (Precalculus). Adding the same amount each time.
Arithmetic Series (Precalculus). Adding up a sequence with a constant step.
Series (Precalculus). Sums of arithmetic and geometric series.

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Pick more than one for a sheet that mixes them.

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Sigma Prep · sigmaprep.io/worksheets Name Sequences Date Period Answer each question. Find a term of an arithmetic sequence Find the 11th term: $1, 0, - 1, - 2, \dots$ Find the 11th term: $- 8, - 11, - 14, - 17, \dots$ Find the 6th term: $2, 3, 4, 5, \dots$ Find the 10th term: $4, 2, 0, - 2, \dots$ Write $a_n$ from the first terms of an arithmetic sequence Find a formula for the nth term: $- 3, - 1, 1, 3, \dots$ Find a formula for the nth term: $6, 7, 8, 9, \dots$ Find a formula for the nth term: $- 9, - 13, - 17, - 21, \dots$ Find a formula for the nth term: $5, 2, - 1, - 4, \dots$ Find a term of a geometric sequence Find the 5th term: $2, 6, 18, 54, \dots$ What is the 6th term of the geometric sequence $- 5, - 10, - 20, - 40, \dots$ ? The sequence $- 32, - 16, - 8, - 4, \dots$ is geometric. Find $a_{5}$ . The sequence $48, 24, 12, 6, \dots$ is geometric. Find $a_{5}$ . Write $a_n$ from two terms of an arithmetic sequence $a_{4} = 19$ and $a_{6} = 25$ . Write the explicit formula for $a_{n}$ $a_{5} = - 3$ and $a_{9} = - 11$ . Write the explicit formula for $a_{n}$ $a_{2} = - 7$ and $a_{6} = 1$ . Write the explicit formula for $a_{n}$ $a_{3} = 2$ and $a_{8} = - 3$ . Write the explicit formula for $a_{n}$ Write $a_n$ from two terms of a geometric sequence In a geometric sequence $a_{3} = 18$ and $a_{4} = - 54$ . Find $a_{n}$ In a geometric sequence $a_{1} = - 2$ and $a_{2} = - 4$ . Find $a_{n}$ In a geometric sequence $a_{3} = - 16$ and $a_{4} = 32$ . Find $a_{n}$ In a geometric sequence $a_{2} = 18$ and $a_{3} = 54$ . Find $a_{n}$ Free math worksheets at sigmaprep.io/worksheets© 2026 Sigma Prep

Sigma Prep · sigmaprep.io/worksheets

Name

Sequences

Date Period

Answer each question.

Find a term of an arithmetic sequence
Find the 11th term: $1, 0, - 1, - 2, \dots$
Find the 11th term: $- 8, - 11, - 14, - 17, \dots$
Find the 6th term: $2, 3, 4, 5, \dots$
Find the 10th term: $4, 2, 0, - 2, \dots$

Write $a_n$ from the first terms of an arithmetic sequence
Find a formula for the nth term: $- 3, - 1, 1, 3, \dots$
Find a formula for the nth term: $6, 7, 8, 9, \dots$
Find a formula for the nth term: $- 9, - 13, - 17, - 21, \dots$
Find a formula for the nth term: $5, 2, - 1, - 4, \dots$

Find a term of a geometric sequence
Find the 5th term: $2, 6, 18, 54, \dots$
What is the 6th term of the geometric sequence $- 5, - 10, - 20, - 40, \dots$ ?
The sequence $- 32, - 16, - 8, - 4, \dots$ is geometric. Find $a_{5}$ .
The sequence $48, 24, 12, 6, \dots$ is geometric. Find $a_{5}$ .

Write $a_n$ from two terms of an arithmetic sequence
$a_{4} = 19$ and $a_{6} = 25$ . Write the explicit formula for $a_{n}$
$a_{5} = - 3$ and $a_{9} = - 11$ . Write the explicit formula for $a_{n}$
$a_{2} = - 7$ and $a_{6} = 1$ . Write the explicit formula for $a_{n}$
$a_{3} = 2$ and $a_{8} = - 3$ . Write the explicit formula for $a_{n}$

Write $a_n$ from two terms of a geometric sequence
In a geometric sequence $a_{3} = 18$ and $a_{4} = - 54$ . Find $a_{n}$
In a geometric sequence $a_{1} = - 2$ and $a_{2} = - 4$ . Find $a_{n}$
In a geometric sequence $a_{3} = - 16$ and $a_{4} = 32$ . Find $a_{n}$
In a geometric sequence $a_{2} = 18$ and $a_{3} = 54$ . Find $a_{n}$



Sigma Prep · sigmaprep.io/worksheets Name SequencesAnswer keyVersion 3 Date Period Answer each question. Find a term of an arithmetic sequence Find the 11th term: $1, 0, - 1, - 2, \dots$ $- 9$ Find the 11th term: $- 8, - 11, - 14, - 17, \dots$ $- 38$ Find the 6th term: $2, 3, 4, 5, \dots$ $7$ Find the 10th term: $4, 2, 0, - 2, \dots$ $- 14$ Write $a_n$ from the first terms of an arithmetic sequence Find a formula for the nth term: $- 3, - 1, 1, 3, \dots$ $a_{n} = 2 n - 5$ Find a formula for the nth term: $6, 7, 8, 9, \dots$ $a_{n} = n + 5$ Find a formula for the nth term: $- 9, - 13, - 17, - 21, \dots$ $a_{n} = - 4 n - 5$ Find a formula for the nth term: $5, 2, - 1, - 4, \dots$ $a_{n} = - 3 n + 8$ Find a term of a geometric sequence Find the 5th term: $2, 6, 18, 54, \dots$ $162$ What is the 6th term of the geometric sequence $- 5, - 10, - 20, - 40, \dots$ ? $- 160$ The sequence $- 32, - 16, - 8, - 4, \dots$ is geometric. Find $a_{5}$ . $- 2$ The sequence $48, 24, 12, 6, \dots$ is geometric. Find $a_{5}$ . $3$ Write $a_n$ from two terms of an arithmetic sequence $a_{4} = 19$ and $a_{6} = 25$ . Write the explicit formula for $a_{n}$ $a_{n} = 3 n + 7$ $a_{5} = - 3$ and $a_{9} = - 11$ . Write the explicit formula for $a_{n}$ $a_{n} = - 2 n + 7$ $a_{2} = - 7$ and $a_{6} = 1$ . Write the explicit formula for $a_{n}$ $a_{n} = 2 n - 11$ $a_{3} = 2$ and $a_{8} = - 3$ . Write the explicit formula for $a_{n}$ $a_{n} = - n + 5$ Write $a_n$ from two terms of a geometric sequence In a geometric sequence $a_{3} = 18$ and $a_{4} = - 54$ . Find $a_{n}$ $a_{n} = 2 \cdot {(- 3)}^{n - 1}$ In a geometric sequence $a_{1} = - 2$ and $a_{2} = - 4$ . Find $a_{n}$ $a_{n} = - 2 \cdot 2^{n - 1}$ In a geometric sequence $a_{3} = - 16$ and $a_{4} = 32$ . Find $a_{n}$ $a_{n} = - 4 \cdot {(- 2)}^{n - 1}$ In a geometric sequence $a_{2} = 18$ and $a_{3} = 54$ . Find $a_{n}$ $a_{n} = 6 \cdot 3^{n - 1}$ Free math worksheets at sigmaprep.io/worksheets© 2026 Sigma Prep

Sigma Prep · sigmaprep.io/worksheets

Name

SequencesAnswer keyVersion 3

Date Period

Answer each question.

Find a term of an arithmetic sequence
Find the 11th term: $1, 0, - 1, - 2, \dots$ $- 9$
Find the 11th term: $- 8, - 11, - 14, - 17, \dots$ $- 38$
Find the 6th term: $2, 3, 4, 5, \dots$ $7$
Find the 10th term: $4, 2, 0, - 2, \dots$ $- 14$

Write $a_n$ from the first terms of an arithmetic sequence
Find a formula for the nth term: $- 3, - 1, 1, 3, \dots$ $a_{n} = 2 n - 5$
Find a formula for the nth term: $6, 7, 8, 9, \dots$ $a_{n} = n + 5$
Find a formula for the nth term: $- 9, - 13, - 17, - 21, \dots$ $a_{n} = - 4 n - 5$
Find a formula for the nth term: $5, 2, - 1, - 4, \dots$ $a_{n} = - 3 n + 8$

Find a term of a geometric sequence
Find the 5th term: $2, 6, 18, 54, \dots$ $162$
What is the 6th term of the geometric sequence $- 5, - 10, - 20, - 40, \dots$ ? $- 160$
The sequence $- 32, - 16, - 8, - 4, \dots$ is geometric. Find $a_{5}$ . $- 2$
The sequence $48, 24, 12, 6, \dots$ is geometric. Find $a_{5}$ . $3$

Write $a_n$ from two terms of an arithmetic sequence
$a_{4} = 19$ and $a_{6} = 25$ . Write the explicit formula for $a_{n}$ $a_{n} = 3 n + 7$
$a_{5} = - 3$ and $a_{9} = - 11$ . Write the explicit formula for $a_{n}$ $a_{n} = - 2 n + 7$
$a_{2} = - 7$ and $a_{6} = 1$ . Write the explicit formula for $a_{n}$ $a_{n} = 2 n - 11$
$a_{3} = 2$ and $a_{8} = - 3$ . Write the explicit formula for $a_{n}$ $a_{n} = - n + 5$

Write $a_n$ from two terms of a geometric sequence
In a geometric sequence $a_{3} = 18$ and $a_{4} = - 54$ . Find $a_{n}$ $a_{n} = 2 \cdot {(- 3)}^{n - 1}$
In a geometric sequence $a_{1} = - 2$ and $a_{2} = - 4$ . Find $a_{n}$ $a_{n} = - 2 \cdot 2^{n - 1}$
In a geometric sequence $a_{3} = - 16$ and $a_{4} = 32$ . Find $a_{n}$ $a_{n} = - 4 \cdot {(- 2)}^{n - 1}$
In a geometric sequence $a_{2} = 18$ and $a_{3} = 54$ . Find $a_{n}$ $a_{n} = 6 \cdot 3^{n - 1}$

How to do these

A geometric sequence has $a_{2} = 6$ and $a_{5} = - 162$ . Find $a_{n}$

From a₂ to a₅ is three steps, so r³ = -162/6 = -27 and r = -3.
a₁ = a₂ ÷ r = 6 ÷ -3 = -2.
a_n = a₁ r^(n - 1).

The answer is $a_{n} = - 2 \cdot {(- 3)}^{n - 1}$ .

Where students go wrong

Leaving out the parentheses around a negative ratio. -2 · -3^(n - 1) raises only the 3 to the power, so every term has the same sign and the alternation is lost.

Also called the nth term, explicit formulas or number patterns.

What you can put on this worksheet

A given term of an arithmetic sequence: Find the 9th term: $6, 7, 8, 9, \dots$ → $14$
A formula for an arithmetic sequence: Find a formula for the nth term: $6, 7, 8, 9, \dots$ → $a_{n} = n + 5$
A given term of a geometric sequence: Find the 6th term: $64, 32, 16, 8, \dots$ → $2$
The formula from two terms: $a_{2} = 10$ and $a_{7} = 25$ . Write the explicit formula for $a_{n}$ → $a_{n} = 3 n + 4$
The explicit formula: Write $a_{n}$ for the geometric sequence whose first term is $- 1$ and whose ratio is $4$ . → $a_{n} = - 4^{n - 1}$
From two terms: In a geometric sequence $a_{1} = 4$ and $a_{2} = - 8$ . Find $a_{n}$ → $a_{n} = 4 \cdot {(- 2)}^{n - 1}$

Questions about these worksheets

Yes. Every arithmetic and geometric sequences sheet is free to print and free to download as a PDF. There is no account to make, no email to hand over and no limit on how many you take.

Yes. The answer key prints on a second page, and you can choose whether it shows each question beside its answer or just the answers in a list.

Yes. Print straight from the page, or download the PDF and print that. The sheet is laid out for paper rather than squeezed off a screen, so there is room to work under each question, and the answer key comes out on its own page.

Precalculus is usually taken around 11th or 12th grade, though schools vary and plenty of students meet it earlier or later. Pick the difficulty rather than the grade: the easy level suits a first lesson on it, the hard level suits review before a test.

Yes, as many as you like. The questions are built fresh each time rather than picked from a fixed set of files, so pressing Generate gives a new sheet. That is what you want for a second class, a retake, or two students sitting next to each other.

That is the closest comparison, and Kuta Software is good software. It is also paid software you install on a computer. These worksheets run in a browser tab for free, with no account and nothing to install. Like Kuta, the arithmetic and geometric sequences questions are generated when you ask for them rather than pulled from a fixed set of files, so two classes never get the same sheet, and the answer key comes with it.

Yes. There are two levels, and you can tick more than one for a sheet that mixes them, in which case the questions come out easiest first. The harder levels are not just larger numbers: they ask for something the easy ones do not.

You pick from 6: a given term of an arithmetic sequence, a formula for an arithmetic sequence, a given term of a geometric sequence, the formula from two terms, the explicit formula and from two terms. Tick as many as you want and set how many of each, or let it spread them evenly.

Multiplying by the term number instead of one less than it. Getting to the 20th term takes 19 steps, not 20, which is the off-by-one that costs most of the marks on this topic.

Arithmetic and geometric sequences worksheet for Precalculus

Select your difficulty

Find a term of an arithmetic sequence

Write ana_nan​ from the first terms of an arithmetic sequence

Find a term of a geometric sequence

Write ana_nan​ from two terms of an arithmetic sequence

Write ana_nan​ from two terms of a geometric sequence

Find a term of an arithmetic sequence

Write ana_nan​ from the first terms of an arithmetic sequence

Find a term of a geometric sequence

Write ana_nan​ from two terms of an arithmetic sequence

Write ana_nan​ from two terms of a geometric sequence

How to do these

Where students go wrong

What you can put on this worksheet

Questions about these worksheets

Are these arithmetic and geometric sequences worksheets free?+

Do the arithmetic and geometric sequences worksheets come with an answer key?+

Are these arithmetic and geometric sequences worksheets printable?+

What grade are these arithmetic and geometric sequences worksheets for?+

Can I get a different version of the same arithmetic and geometric sequences worksheet?+

Is this a free alternative to Kuta Software for arithmetic and geometric sequences?+

Can I choose how hard the arithmetic and geometric sequences questions are?+

What kinds of question are on a arithmetic and geometric sequences worksheet?+

What do students get wrong with arithmetic and geometric sequences?+

Write $a_n$ from the first terms of an arithmetic sequence

Write $a_n$ from two terms of an arithmetic sequence

Write $a_n$ from two terms of a geometric sequence

Write $a_n$ from the first terms of an arithmetic sequence

Write $a_n$ from two terms of an arithmetic sequence

Write $a_n$ from two terms of a geometric sequence