Sequences and series worksheet for Precalculus

Precalculus treats a sequence as a function on the counting numbers and uses sigma notation in both directions: evaluating a sum, and writing a list as one. The patterns can be arithmetic, geometric, or something else like n² + 1.

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

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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 and Series Date Period Answer each question. Find a term from the formula Given $a_{n} = 12 n - 21$ , find $a_{16}$ Given $a_{n} = - 11 n - 25$ , find $a_{16}$ Given $a_{n} = - 15 n^{2} + 20$ , find $a_{6}$ Given $a_{n} = - 11 n + 16$ , find $a_{19}$ Given $a_{n} = 14 n + 20$ , find $a_{7}$ Evaluate the sum Evaluate: $\sum_{k = 4}^{8} (- 10 k + 13)$ $\sum_{k = 1}^{5} (9 k - 15)$ $\sum_{k = 1}^{8} (- 9 k + 17)$ Evaluate: $\sum_{k = 3}^{5} (10 k - 18)$ $\sum_{k = 1}^{5} (- 9 k - 19)$ Write in sigma notation Write in sigma notation: $4 + 8 + 16 + \dots + 256$ Write in sigma notation: $- 5 - 1 + 3 + \dots + 79$ Write in sigma notation: $2 + 6 + 18 + 54$ Write in sigma notation: $- 3 + 2 + 7 + \dots + 77$ Write in sigma notation: $0 - 6 - 12 - \dots - 66$ Write a rule for the $n$ th term Write a rule for the nth term: $- 1, 2, - 3, 4, \dots$ Write a rule for the nth term: $\frac{1}{3}, \frac{4}{5}, \frac{9}{7}, \frac{16}{9}, \dots$ Write a rule for the nth term: $1, - 3, 5, - 7, \dots$ Write a rule for the nth term: $1, \frac{3}{4}, \frac{5}{9}, \frac{7}{16}, \dots$ Write a rule for the nth term: $\frac{1}{2}, - \frac{1}{4}, \frac{1}{8}, - \frac{1}{16}, \dots$ Free math worksheets at sigmaprep.io/worksheets© 2026 Sigma Prep

Sigma Prep · sigmaprep.io/worksheets

Name

Sequences and Series

Date Period

Answer each question.

Find a term from the formula
Given $a_{n} = 12 n - 21$ , find $a_{16}$
Given $a_{n} = - 11 n - 25$ , find $a_{16}$
Given $a_{n} = - 15 n^{2} + 20$ , find $a_{6}$
Given $a_{n} = - 11 n + 16$ , find $a_{19}$
Given $a_{n} = 14 n + 20$ , find $a_{7}$

Evaluate the sum
Evaluate: $\sum_{k = 4}^{8} (- 10 k + 13)$
$\sum_{k = 1}^{5} (9 k - 15)$
$\sum_{k = 1}^{8} (- 9 k + 17)$
Evaluate: $\sum_{k = 3}^{5} (10 k - 18)$
$\sum_{k = 1}^{5} (- 9 k - 19)$

Write in sigma notation
Write in sigma notation: $4 + 8 + 16 + \dots + 256$
Write in sigma notation: $- 5 - 1 + 3 + \dots + 79$
Write in sigma notation: $2 + 6 + 18 + 54$
Write in sigma notation: $- 3 + 2 + 7 + \dots + 77$
Write in sigma notation: $0 - 6 - 12 - \dots - 66$

Write a rule for the $n$ th term
Write a rule for the nth term: $- 1, 2, - 3, 4, \dots$
Write a rule for the nth term:
$\frac{1}{3}, \frac{4}{5}, \frac{9}{7}, \frac{16}{9}, \dots$
Write a rule for the nth term: $1, - 3, 5, - 7, \dots$
Write a rule for the nth term:
$1, \frac{3}{4}, \frac{5}{9}, \frac{7}{16}, \dots$
Write a rule for the nth term:
$\frac{1}{2}, - \frac{1}{4}, \frac{1}{8}, - \frac{1}{16}, \dots$



Sigma Prep · sigmaprep.io/worksheets Name Sequences and SeriesAnswer keyVersion 2 Date Period Answer each question. Find a term from the formula Given $a_{n} = 12 n - 21$ , find $a_{16}$ $171$ Given $a_{n} = - 11 n - 25$ , find $a_{16}$ $- 201$ Given $a_{n} = - 15 n^{2} + 20$ , find $a_{6}$ $- 520$ Given $a_{n} = - 11 n + 16$ , find $a_{19}$ $- 193$ Given $a_{n} = 14 n + 20$ , find $a_{7}$ $118$ Evaluate the sum Evaluate: $\sum_{k = 4}^{8} (- 10 k + 13)$ $- 235$ $\sum_{k = 1}^{5} (9 k - 15)$ $60$ $\sum_{k = 1}^{8} (- 9 k + 17)$ $- 188$ Evaluate: $\sum_{k = 3}^{5} (10 k - 18)$ $66$ $\sum_{k = 1}^{5} (- 9 k - 19)$ $- 230$ Write in sigma notation Write in sigma notation: $4 + 8 + 16 + \dots + 256$ $\sum_{n = 1}^{7} 4 \cdot 2^{n - 1}$ Write in sigma notation: $- 5 - 1 + 3 + \dots + 79$ $\sum_{n = 1}^{22} (4 n - 9)$ Write in sigma notation: $2 + 6 + 18 + 54$ $\sum_{n = 1}^{4} 2 \cdot 3^{n - 1}$ Write in sigma notation: $- 3 + 2 + 7 + \dots + 77$ $\sum_{n = 1}^{17} (5 n - 8)$ Write in sigma notation: $0 - 6 - 12 - \dots - 66$ $\sum_{n = 1}^{12} (- 6 n + 6)$ Write a rule for the $n$ th term Write a rule for the nth term: $- 1, 2, - 3, 4, \dots$ $a_{n} = {(- 1)}^{n} n$ Write a rule for the nth term: $\frac{1}{3}, \frac{4}{5}, \frac{9}{7}, \frac{16}{9}, \dots$ $a_{n} = \frac{n^{2}}{2 n + 1}$ Write a rule for the nth term: $1, - 3, 5, - 7, \dots$ $a_{n} = {(- 1)}^{n + 1} (2 n - 1)$ Write a rule for the nth term: $1, \frac{3}{4}, \frac{5}{9}, \frac{7}{16}, \dots$ $a_{n} = \frac{2 n - 1}{n^{2}}$ Write a rule for the nth term: $\frac{1}{2}, - \frac{1}{4}, \frac{1}{8}, - \frac{1}{16}, \dots$ $a_{n} = \frac{{(- 1)}^{n + 1}}{2^{n}}$ Free math worksheets at sigmaprep.io/worksheets© 2026 Sigma Prep

Sigma Prep · sigmaprep.io/worksheets

Name

Sequences and SeriesAnswer keyVersion 2

Date Period

Answer each question.

Find a term from the formula
Given $a_{n} = 12 n - 21$ , find $a_{16}$ $171$
Given $a_{n} = - 11 n - 25$ , find $a_{16}$ $- 201$
Given $a_{n} = - 15 n^{2} + 20$ , find $a_{6}$ $- 520$
Given $a_{n} = - 11 n + 16$ , find $a_{19}$ $- 193$
Given $a_{n} = 14 n + 20$ , find $a_{7}$ $118$

Evaluate the sum
Evaluate: $\sum_{k = 4}^{8} (- 10 k + 13)$ $- 235$
$\sum_{k = 1}^{5} (9 k - 15)$ $60$
$\sum_{k = 1}^{8} (- 9 k + 17)$ $- 188$
Evaluate: $\sum_{k = 3}^{5} (10 k - 18)$ $66$
$\sum_{k = 1}^{5} (- 9 k - 19)$ $- 230$

Write in sigma notation
Write in sigma notation: $4 + 8 + 16 + \dots + 256$ $\sum_{n = 1}^{7} 4 \cdot 2^{n - 1}$
Write in sigma notation: $- 5 - 1 + 3 + \dots + 79$ $\sum_{n = 1}^{22} (4 n - 9)$
Write in sigma notation: $2 + 6 + 18 + 54$ $\sum_{n = 1}^{4} 2 \cdot 3^{n - 1}$
Write in sigma notation: $- 3 + 2 + 7 + \dots + 77$ $\sum_{n = 1}^{17} (5 n - 8)$
Write in sigma notation: $0 - 6 - 12 - \dots - 66$ $\sum_{n = 1}^{12} (- 6 n + 6)$

Write a rule for the $n$ th term
Write a rule for the nth term: $- 1, 2, - 3, 4, \dots$ $a_{n} = {(- 1)}^{n} n$
Write a rule for the nth term:
$\frac{1}{3}, \frac{4}{5}, \frac{9}{7}, \frac{16}{9}, \dots$ $a_{n} = \frac{n^{2}}{2 n + 1}$
Write a rule for the nth term: $1, - 3, 5, - 7, \dots$ $a_{n} = {(- 1)}^{n + 1} (2 n - 1)$
Write a rule for the nth term:
$1, \frac{3}{4}, \frac{5}{9}, \frac{7}{16}, \dots$ $a_{n} = \frac{2 n - 1}{n^{2}}$
Write a rule for the nth term:
$\frac{1}{2}, - \frac{1}{4}, \frac{1}{8}, - \frac{1}{16}, \dots$ $a_{n} = \frac{{(- 1)}^{n + 1}}{2^{n}}$

How to do these

Write in sigma notation: $3 + 7 + 11 + \dots + 39$

The terms go up by 4, so the nth term is 4n - 1.
Solve 4n - 1 = 39 to get n = 10.
So n runs from 1 to 10.

The answer is $\sum_{n = 1}^{10} (4 n - 1)$ .

Where students go wrong

Putting the last term on top of the sigma. In 3 + 7 + ... + 39 the top limit is 10, the number of terms, not 39, the value of the last one.

Also called nth term, sigma notation or summation notation.

What you can put on this worksheet

A term from an explicit formula: Given $a_{n} = 4 n$ , find $a_{5}$ → $20$
Add up a short series: $\sum_{k = 1}^{4} (3 k)$ → $30$
Write in sigma notation: Write in sigma notation: $8 + 12 + 16 + \dots + 28$ → $\sum_{n = 1}^{6} (4 n + 4)$
Other patterns: Write a rule for the nth term: $2, 6, 12, 20, \dots$ → $a_{n} = n (n + 1)$
Patterns, far terms, writing out, evaluating and shifting sigma: Write out the sum: $\sum_{m = 1}^{6} ⁣ (44 - m^{2})$ → $43 + 40 + 35 + 28 + 19 + 8$

Questions about these worksheets

Yes. Every sequences and series 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 sequences and series 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 three 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 5: a term from an explicit formula, add up a short series, write in sigma notation, other patterns and patterns, far terms, writing out, evaluating and shifting sigma. Tick as many as you want and set how many of each, or let it spread them evenly.

Miscounting the terms in a sum. The sum from k = 4 to 6 has three terms, k = 4, 5 and 6, not six: the index starts where the bottom says, not at 1.

Sequences and series worksheet for Precalculus

Select your difficulty

Find a term from the formula

Evaluate the sum

Write in sigma notation

Write a rule for the nnnth term

Find a term from the formula

Evaluate the sum

Write in sigma notation

Write a rule for the nnnth term

How to do these

Where students go wrong

What you can put on this worksheet

Questions about these worksheets

Are these sequences and series worksheets free?+

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

Are these sequences and series worksheets printable?+

What grade are these sequences and series worksheets for?+

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

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

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

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

What do students get wrong with sequences and series?+

Write a rule for the $n$ th term

Write a rule for the $n$ th term