Worksheets · Algebra 1

Recursive sequences worksheet for Algebra 1

Algebra 1 builds sequences one step at a time: a first term, then a rule such as add 5 or multiply by -2. You write the rule from a list, step forward to find terms, and turn an arithmetic rule into the explicit form a_n = a_1 + (n - 1)d.

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

Select your difficulty

Pick more than one for a sheet that mixes them.

The preview can look cramped on a phone. The PDF and printed sheet come out normal.

Free math worksheets at sigmaprep.io/worksheets© 2026 Sigma Prep

Sigma Prep · sigmaprep.io/worksheets

Name

Recursive Sequences

Date Period

Find each term, or write the rule.

  1. Find a term from a recursive rule

    A sequence has a1=2 and an=an−1+4. Find a4
  2. a1=3, a2=2, an=2an−1+an−2. Find a7
  3. A sequence has a1=1 and an=2an−1. Find a4
  4. a1=3, a2=4, an=an−1−an−2. Find a5
  5. A sequence has a1=4 and an=an−1−5. Find a4
  6. a1=−3, an=−2an−1+3. Find a4
  7. A sequence has a1=6 and an=an−1−3. Find a4
  8. a1=−1, an=3an−1+4. Find a4
  9. A sequence has a1=4 and an=an−1+4. Find a4
  10. a1=2, an=2an−1−5. Find a4

  1. Write the recursive rule

    Write a recursive rule: −4,−3,−2,−1,…
  2. Write a recursive rule: −6,−4,−2,0,…
  3. Write a recursive rule: 8,0,−8,−16,…
  4. Write a recursive rule: 2,4,8,16,…
  5. Write a recursive rule: −8,16,−32,64,…

  1. Convert between explicit and recursive rules

    Write an explicit rule for a1=−4, an=an−1+6
  2. Write a recursive rule for an=n−2
  3. Write a recursive rule for an=6n+3
  4. Write a recursive rule for an=−5n−1
  5. Write an explicit rule for a1=4, an=an−1+8

Free math worksheets at sigmaprep.io/worksheets© 2026 Sigma Prep

Sigma Prep · sigmaprep.io/worksheets

Name

Recursive SequencesAnswer keyVersion 2

Date Period

Find each term, or write the rule.

  1. Find a term from a recursive rule

    A sequence has a1=2 and an=an−1+4. Find a414
  2. a1=3, a2=2, an=2an−1+an−2. Find a7227
  3. A sequence has a1=1 and an=2an−1. Find a48
  4. a1=3, a2=4, an=an−1−an−2. Find a5−4
  5. A sequence has a1=4 and an=an−1−5. Find a4−11
  6. a1=−3, an=−2an−1+3. Find a433
  7. A sequence has a1=6 and an=an−1−3. Find a4−3
  8. a1=−1, an=3an−1+4. Find a425
  9. A sequence has a1=4 and an=an−1+4. Find a416
  10. a1=2, an=2an−1−5. Find a4−19

  1. Write the recursive rule

    Write a recursive rule: −4,−3,−2,−1,…a1=−4, an=an−1+1
  2. Write a recursive rule: −6,−4,−2,0,…a1=−6, an=an−1+2
  3. Write a recursive rule: 8,0,−8,−16,…a1=8, an=an−1−8
  4. Write a recursive rule: 2,4,8,16,…a1=2, an=2an−1
  5. Write a recursive rule: −8,16,−32,64,…a1=−8, an=−2an−1

  1. Convert between explicit and recursive rules

    Write an explicit rule for a1=−4, an=an−1+6an=6n−10
  2. Write a recursive rule for an=n−2a1=−1, an=an−1+1
  3. Write a recursive rule for an=6n+3a1=9, an=an−1+6
  4. Write a recursive rule for an=−5n−1a1=−6, an=an−1−5
  5. Write an explicit rule for a1=4, an=an−1+8an=8n−4

Free math worksheets at sigmaprep.io/worksheets© 2026 Sigma Prep

How to do these

Write a recursive rule: 5,8,11,14,…

  1. The first term is 5.
  2. Each term is 3 more than the one before.

The answer is a1=5, an=an−1+3.

Where students go wrong

Writing a recursive rule without a first term. a_n = a_(n-1) + 3 fits every sequence that goes up by 3, so it needs a₁ to pick out the right one.

Also called recursive rule, recursive formula or term from the previous term.

What you can put on this worksheet

A term from a recursive rule
A sequence has a1=4 and an=an−1+2. Find a5 → 12
Write the recursive rule
Write a recursive rule: 6,7,8,9,… → a1=6, an=an−1+1
Between explicit and recursive
Write an explicit rule for a1=7, an=an−1−5 → an=−5n+12
Mixed and two-term rules
a1=1, an=3an−1+1. Find a4 → 40
From far terms, rules with n or a fraction, growing gaps
Write a recursive formula: 5,6,8,11,15,… → an=an−1+n−1a1=5

Questions about these worksheets

Yes. Every recursive 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.

Algebra 1 is usually taken around 8th or 9th 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 recursive 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 5: a term from a recursive rule, write the recursive rule, between explicit and recursive, mixed and two-term rules and from far terms, rules with n or a fraction, growing gaps. Tick as many as you want and set how many of each, or let it spread them evenly.

Taking as many steps as the term number. Getting to the 5th term from the 1st is four steps, not five.