Find the first five terms and a_23: a_n = 1/2 + 1/3n

5/6, 7/6, 3/2, 11/6, 13/6 a_23 = 49/6

a_2 = 14 and a_17 = 74. Find d, an explicit rule and a recursive rule.

d = 4; a_n = 4n + 6;; a_1 = 10, a_n = a_n-1 + 4 for n ≥ 2

Arithmetic sequences worksheet for Precalculus

Precalculus treats an arithmetic sequence as a linear function whose inputs are the counting numbers, written with subscripts. Much of the work runs backwards: building the explicit formula from two scattered terms or from a list, and finding which term a given value is.

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

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

Sequences (Precalculus). A term, which term, or a formula for the nth term.
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 Arithmetic Sequences Date Period Answer each question. Find a term of an arithmetic sequence Find the 12th term: $4, 0, - 4, - 8, \dots$ Find the 12th term: $4, 5, 6, 7, \dots$ Find the 12th term: $- 6, - 7, - 8, - 9, \dots$ Find the 8th term: $- 8, - 4, 0, 4, \dots$ Find the 12th term: $4, - 1, - 6, - 11, \dots$ Find the 10th term: $- 6, - 4, - 2, 0, \dots$ Find the 11th term: $- 5, 0, 5, 10, \dots$ Write $a_n$ from the first terms of an arithmetic sequence Find a formula for the nth term: $- 8, - 13, - 18, - 23, \dots$ Find a formula for the nth term: $- 1, 2, 5, 8, \dots$ Find a formula for the nth term: $1, 5, 9, 13, \dots$ Find a formula for the nth term: $- 6, - 1, 4, 9, \dots$ Find a formula for the nth term: $2, 1, 0, - 1, \dots$ Find a formula for the nth term: $6, 0, - 6, - 12, \dots$ Find a formula for the nth term: $- 5, - 3, - 1, 1, \dots$ Write $a_n$ from two terms of an arithmetic sequence $a_{4} = 1$ and $a_{6} = - 5$ . Write the explicit formula for $a_{n}$ $a_{4} = 7$ and $a_{8} = 27$ . Write the explicit formula for $a_{n}$ $a_{5} = - 30$ and $a_{13} = - 62$ . Write the explicit formula for $a_{n}$ $a_{3} = - 7$ and $a_{10} = 7$ . Write the explicit formula for $a_{n}$ $a_{5} = 19$ and $a_{9} = 35$ . Write the explicit formula for $a_{n}$ $a_{5} = - 15$ and $a_{7} = - 21$ . Write the explicit formula for $a_{n}$ Free math worksheets at sigmaprep.io/worksheets© 2026 Sigma Prep

Sigma Prep · sigmaprep.io/worksheets

Name

Arithmetic Sequences

Date Period

Answer each question.

Find a term of an arithmetic sequence
Find the 12th term: $4, 0, - 4, - 8, \dots$
Find the 12th term: $4, 5, 6, 7, \dots$
Find the 12th term: $- 6, - 7, - 8, - 9, \dots$
Find the 8th term: $- 8, - 4, 0, 4, \dots$
Find the 12th term: $4, - 1, - 6, - 11, \dots$
Find the 10th term: $- 6, - 4, - 2, 0, \dots$
Find the 11th term: $- 5, 0, 5, 10, \dots$

Write $a_n$ from the first terms of an arithmetic sequence
Find a formula for the nth term: $- 8, - 13, - 18, - 23, \dots$
Find a formula for the nth term: $- 1, 2, 5, 8, \dots$
Find a formula for the nth term: $1, 5, 9, 13, \dots$
Find a formula for the nth term: $- 6, - 1, 4, 9, \dots$
Find a formula for the nth term: $2, 1, 0, - 1, \dots$
Find a formula for the nth term: $6, 0, - 6, - 12, \dots$
Find a formula for the nth term: $- 5, - 3, - 1, 1, \dots$

Write $a_n$ from two terms of an arithmetic sequence
$a_{4} = 1$ and $a_{6} = - 5$ . Write the explicit formula for $a_{n}$
$a_{4} = 7$ and $a_{8} = 27$ . Write the explicit formula for $a_{n}$
$a_{5} = - 30$ and $a_{13} = - 62$ . Write the explicit formula for $a_{n}$
$a_{3} = - 7$ and $a_{10} = 7$ . Write the explicit formula for $a_{n}$
$a_{5} = 19$ and $a_{9} = 35$ . Write the explicit formula for $a_{n}$
$a_{5} = - 15$ and $a_{7} = - 21$ . Write the explicit formula for $a_{n}$



Sigma Prep · sigmaprep.io/worksheets Name Arithmetic SequencesAnswer keyVersion 2 Date Period Answer each question. Find a term of an arithmetic sequence Find the 12th term: $4, 0, - 4, - 8, \dots$ $- 40$ Find the 12th term: $4, 5, 6, 7, \dots$ $15$ Find the 12th term: $- 6, - 7, - 8, - 9, \dots$ $- 17$ Find the 8th term: $- 8, - 4, 0, 4, \dots$ $20$ Find the 12th term: $4, - 1, - 6, - 11, \dots$ $- 51$ Find the 10th term: $- 6, - 4, - 2, 0, \dots$ $12$ Find the 11th term: $- 5, 0, 5, 10, \dots$ $45$ Write $a_n$ from the first terms of an arithmetic sequence Find a formula for the nth term: $- 8, - 13, - 18, - 23, \dots$ $a_{n} = - 5 n - 3$ Find a formula for the nth term: $- 1, 2, 5, 8, \dots$ $a_{n} = 3 n - 4$ Find a formula for the nth term: $1, 5, 9, 13, \dots$ $a_{n} = 4 n - 3$ Find a formula for the nth term: $- 6, - 1, 4, 9, \dots$ $a_{n} = 5 n - 11$ Find a formula for the nth term: $2, 1, 0, - 1, \dots$ $a_{n} = - n + 3$ Find a formula for the nth term: $6, 0, - 6, - 12, \dots$ $a_{n} = - 6 n + 12$ Find a formula for the nth term: $- 5, - 3, - 1, 1, \dots$ $a_{n} = 2 n - 7$ Write $a_n$ from two terms of an arithmetic sequence $a_{4} = 1$ and $a_{6} = - 5$ . Write the explicit formula for $a_{n}$ $a_{n} = - 3 n + 13$ $a_{4} = 7$ and $a_{8} = 27$ . Write the explicit formula for $a_{n}$ $a_{n} = 5 n - 13$ $a_{5} = - 30$ and $a_{13} = - 62$ . Write the explicit formula for $a_{n}$ $a_{n} = - 4 n - 10$ $a_{3} = - 7$ and $a_{10} = 7$ . Write the explicit formula for $a_{n}$ $a_{n} = 2 n - 13$ $a_{5} = 19$ and $a_{9} = 35$ . Write the explicit formula for $a_{n}$ $a_{n} = 4 n - 1$ $a_{5} = - 15$ and $a_{7} = - 21$ . Write the explicit formula for $a_{n}$ $a_{n} = - 3 n$ Free math worksheets at sigmaprep.io/worksheets© 2026 Sigma Prep

Sigma Prep · sigmaprep.io/worksheets

Name

Arithmetic SequencesAnswer keyVersion 2

Date Period

Answer each question.

Find a term of an arithmetic sequence
Find the 12th term: $4, 0, - 4, - 8, \dots$ $- 40$
Find the 12th term: $4, 5, 6, 7, \dots$ $15$
Find the 12th term: $- 6, - 7, - 8, - 9, \dots$ $- 17$
Find the 8th term: $- 8, - 4, 0, 4, \dots$ $20$
Find the 12th term: $4, - 1, - 6, - 11, \dots$ $- 51$
Find the 10th term: $- 6, - 4, - 2, 0, \dots$ $12$
Find the 11th term: $- 5, 0, 5, 10, \dots$ $45$

Write $a_n$ from the first terms of an arithmetic sequence
Find a formula for the nth term: $- 8, - 13, - 18, - 23, \dots$ $a_{n} = - 5 n - 3$
Find a formula for the nth term: $- 1, 2, 5, 8, \dots$ $a_{n} = 3 n - 4$
Find a formula for the nth term: $1, 5, 9, 13, \dots$ $a_{n} = 4 n - 3$
Find a formula for the nth term: $- 6, - 1, 4, 9, \dots$ $a_{n} = 5 n - 11$
Find a formula for the nth term: $2, 1, 0, - 1, \dots$ $a_{n} = - n + 3$
Find a formula for the nth term: $6, 0, - 6, - 12, \dots$ $a_{n} = - 6 n + 12$
Find a formula for the nth term: $- 5, - 3, - 1, 1, \dots$ $a_{n} = 2 n - 7$

Write $a_n$ from two terms of an arithmetic sequence
$a_{4} = 1$ and $a_{6} = - 5$ . Write the explicit formula for $a_{n}$ $a_{n} = - 3 n + 13$
$a_{4} = 7$ and $a_{8} = 27$ . Write the explicit formula for $a_{n}$ $a_{n} = 5 n - 13$
$a_{5} = - 30$ and $a_{13} = - 62$ . Write the explicit formula for $a_{n}$ $a_{n} = - 4 n - 10$
$a_{3} = - 7$ and $a_{10} = 7$ . Write the explicit formula for $a_{n}$ $a_{n} = 2 n - 13$
$a_{5} = 19$ and $a_{9} = 35$ . Write the explicit formula for $a_{n}$ $a_{n} = 4 n - 1$
$a_{5} = - 15$ and $a_{7} = - 21$ . Write the explicit formula for $a_{n}$ $a_{n} = - 3 n$

How to do these

$a_{5} = 17$ and $a_{12} = 52$ . Write the explicit formula for $a_{n}$

From a₅ to a₁₂ is 7 steps, and the value rises by 35, so d = 5.
Step back 4 times from a₅: a₁ = 17 - 20 = -3.
a_n = -3 + 5(n - 1), which simplifies to 5n - 8.

The answer is $a_{n} = 5 n - 8$ .

Where students go wrong

Dividing by the wrong gap. From a₅ to a₁₂ is 12 - 5 = 7 steps, so the change of 35 is shared over 7, not over 12 or 17.

Also called common difference, nth term of an arithmetic sequence or explicit formula.

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: $6, 7, 8, 9, \dots$ → $a_{n} = n + 5$
Insert arithmetic means: Insert 2 arithmetic means between $1$ and $- 8$ → $- 2, - 5$
The formula from two terms: $a_{2} = 10$ and $a_{7} = 25$ . Write the explicit formula for $a_{n}$ → $a_{n} = 3 n + 4$
Means, decimal and fraction formulas, working back from a far term: Find the first five terms and $a_{23}$ : $a_{n} = \frac{1}{2} + \frac{1}{3} n$ → $\frac{5}{6}, \frac{7}{6}, \frac{3}{2}, \frac{11}{6}, \frac{13}{6} a_{23} = \frac{49}{6}$
From two far-apart terms: d, explicit and recursive: $a_{2} = 14$ and $a_{17} = 74$ . Find $d$ , an explicit rule and a recursive rule. → $d = 4$ ; $a_{n} = 4 n + 6$ ;
$a_{1} = 10, a_{n} = a_{n - 1} + 4$ for $n \geq 2$

Questions about these worksheets

Yes. Every arithmetic 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 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, insert arithmetic means, the formula from two terms, means, decimal and fraction formulas, working back from a far term and from two far-apart terms: d, explicit and recursive. Tick as many as you want and set how many of each, or let it spread them evenly.

Multiplying by the term number rather than one less. Reaching the 20th term takes 19 steps, not 20.

Arithmetic 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

Write ana_nan​ from two terms of an arithmetic sequence

Find a term of an arithmetic sequence

Write ana_nan​ from the first terms of an arithmetic sequence

Write ana_nan​ from two terms of an arithmetic sequence

How to do these

Where students go wrong

What you can put on this worksheet

Questions about these worksheets

Are these arithmetic sequences worksheets free?+

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

Are these arithmetic sequences worksheets printable?+

What grade are these arithmetic sequences worksheets for?+

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

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

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

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

What do students get wrong with arithmetic 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 the first terms of an arithmetic sequence

Write $a_n$ from two terms of an arithmetic sequence