Worksheets · Precalculus

Precalculus recursive sequences worksheet

In Precalculus recursive rules go past adding or multiplying by the same number each time. Rules like a_n = 3a_{n-1} - 1 mix both, so there is no shortcut and every term comes from the one before it.

Also taught in Algebra 2 and Algebra 1, 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. A sequence has a1=7 and an=an−1+9. Find a5
  2. Write a recursive rule: 7,21,63,189,…
  3. Write a recursive rule for an=−3n+1
  4. a1=2, a2=1, an=an−1−an−2. Find a5
  5. A sequence has a1=7 and an=3an−1. Find a5
  6. Write a recursive rule: −2,3,8,13,…
  7. Write a recursive rule for an=9n−4
  8. a1=3, a2=1, an=an−1+an−2. Find a5
  9. A sequence has a1=9 and an=an−1−9. Find a6
  10. Write a recursive rule: 3,−6,12,−24,…
  11. Write an explicit rule for a1=2, an=an−1+9
  12. a1=2, a2=3, an=2an−1+an−2. Find a7
  13. A sequence has a1=7 and an=an−1+7. Find a4
  14. Write a recursive rule: −2,4,−8,16,…
  15. Write a recursive rule for an=6n+2
  16. a1=2, an=3an−1+5. Find a4
  17. A sequence has a1=9 and an=3an−1. Find a5
  18. Write a recursive rule: −6,−18,−54,−162,…
  19. Write an explicit rule for a1=−5, an=an−1+6
  20. a1=−3, an=2an−1−4. Find a4

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

Sigma Prep · sigmaprep.io/worksheets

Name

Recursive SequencesAnswer keyVersion 1

Date Period

Find each term, or write the rule.

  1. A sequence has a1=7 and an=an−1+9. Find a543
  2. Write a recursive rule: 7,21,63,189,…a1=7, an=3an−1
  3. Write a recursive rule for an=−3n+1a1=−2, an=an−1−3
  4. a1=2, a2=1, an=an−1−an−2. Find a5−1
  5. A sequence has a1=7 and an=3an−1. Find a5567
  6. Write a recursive rule: −2,3,8,13,…a1=−2, an=an−1+5
  7. Write a recursive rule for an=9n−4a1=5, an=an−1+9
  8. a1=3, a2=1, an=an−1+an−2. Find a59
  9. A sequence has a1=9 and an=an−1−9. Find a6−36
  10. Write a recursive rule: 3,−6,12,−24,…a1=3, an=−2an−1
  11. Write an explicit rule for a1=2, an=an−1+9an=9n−7
  12. a1=2, a2=3, an=2an−1+an−2. Find a7268
  13. A sequence has a1=7 and an=an−1+7. Find a428
  14. Write a recursive rule: −2,4,−8,16,…a1=−2, an=−2an−1
  15. Write a recursive rule for an=6n+2a1=8, an=an−1+6
  16. a1=2, an=3an−1+5. Find a4119
  17. A sequence has a1=9 and an=3an−1. Find a5729
  18. Write a recursive rule: −6,−18,−54,−162,…a1=−6, an=3an−1
  19. Write an explicit rule for a1=−5, an=an−1+6an=6n−11
  20. a1=−3, an=2an−1−4. Find a4−52

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

How to do these

a1=2, an=3an−1−1. Find a4

  1. a₂ = 3(2) - 1 = 5.
  2. a₃ = 3(5) - 1 = 14.
  3. a₄ = 3(14) - 1 = 41.

The answer is a4=41.

Where students go wrong

Treating the rule as arithmetic or geometric. The gaps here are 3, 9, 27 and the ratios change too, so build the terms one at a time.

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.

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 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.