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Introduction
Recursive Rules helps you connect a mathematical rule to a result you can explain and check. Supply an initial term and apply a repeated process to generate later terms. Compare arithmetic and geometric recursion. This lesson belongs to sequences, data and mathematical modeling and builds toward independent Algebra I problem solving.
Alignment uses 19 TAC §111.39(c). Review lessons sample their listed standards. Readiness and supporting labels describe STAAR emphasis, not whether a topic is required.
Learning objectives
- Identify the quantities and representations used in recursive rules.
- Apply the lesson's rule to solve a numerical or contextual problem.
- Explain each mathematical step using the correct vocabulary.
- Verify an answer and correct a likely error.
Prerequisite knowledge
Identify repeated differences or ratios and evaluate powers.
Previous lesson: Arithmetic Sequences
Key mathematical vocabulary
- constant
- A value fixed within a problem.
- domain
- The set of permitted inputs.
- sequence
- An ordered list indexed by integers.
- ratio
- A comparison by division.
- rate
- A ratio measured in units per unit.
Formulas and rules
- aₙ=a₁+(n−1)d
- aₙ=a₁rn−1
Arithmetic and geometric sequences
Supply an initial term and apply a repeated process to generate later terms. Compare arithmetic and geometric recursion.
A sequence is an ordered list with a domain of integer indices, usually n≥1. An arithmetic sequence has constant difference d; a geometric sequence has constant ratio r when the required divisions are defined.
With first term a₁, explicit rules are aₙ=a₁+(n−1)d and aₙ=a₁rn−1. Recursive rules are aₙ=aₙ₋₁+d and aₙ=raₙ₋₁ for n≥2, together with the first term. A few terms suggest a model but do not uniquely determine an infinite sequence without a stated pattern assumption.
A method you can explain
- Read the question and identify the known information, unknown quantities, and any domain restrictions. Select a suitable representation and tool.
- Apply the relationship explained above. The first term is 33; the difference is 12.
- Write each calculation or logical inference on a separate line. Keep units and signs visible. When two methods are possible, compare which one uses the given information most directly.
- Check the result in the ORIGINAL conditions, not merely the last simplified line. Undo the n−1 changes to recover the first term.
- Interpret what the answer means. Decide whether its size, sign, and precision make sense; identify a restriction or assumption that would change the conclusion.
Five worked examples
Example 1 · Beginner
Given a₁=33 and aₙ=aₙ₋₁+12 for n≥2, find a6.
| n | aₙ |
|---|---|
| 1 | 33 |
| 2 | 45 |
| 3 | 57 |
| 4 | 69 |
| 5 | 81 |
- The first term is 33; the difference is 12.
- From term 1 to term n there are n−1 steps. Add the common difference at each step.
- At n=6, 33+12·5=93. A recursive rule needs a starting term; an explicit rule calculates any term directly.
Answer: aₙ = 33 + 12(n−1); a6 = 93
Verify: Undo the n−1 changes to recover the first term.
Avoid this mistake: Using n instead of n−1 with indexing starting at 1.
Example 2 · Basic
Given a₁=33 and aₙ=13aₙ₋₁ for n≥2, find a3.
| n | aₙ |
|---|---|
| 1 | 33 |
| 2 | 429 |
| 3 | 5577 |
| 4 | 72501 |
| 5 | 942513 |
- The first term is 33; the ratio is 13.
- From term 1 to term n there are n−1 steps. Multiply by the ratio at each step.
- At n=3, 33·132=5577. A recursive rule needs a starting term; an explicit rule calculates any term directly.
Answer: aₙ = 33·13n−1; a3 = 5577
Verify: Undo the n−1 changes to recover the first term.
Avoid this mistake: Omitting the starting term of a recursive rule.
Example 3 · Intermediate
Given a₁=33 and aₙ=aₙ₋₁+14 for n≥2, find a4.
| n | aₙ |
|---|---|
| 1 | 33 |
| 2 | 47 |
| 3 | 61 |
| 4 | 75 |
| 5 | 89 |
- The first term is 33; the difference is 14.
- From term 1 to term n there are n−1 steps. Add the common difference at each step.
- At n=4, 33+14·3=75. A recursive rule needs a starting term; an explicit rule calculates any term directly.
Answer: aₙ = 33 + 14(n−1); a4 = 75
Verify: Undo the n−1 changes to recover the first term.
Avoid this mistake: Confusing a common difference with a common ratio.
Example 4 · Advanced
Given a₁=17 and aₙ=2aₙ₋₁ for n≥2, find a5.
| n | aₙ |
|---|---|
| 1 | 17 |
| 2 | 34 |
| 3 | 68 |
| 4 | 136 |
| 5 | 272 |
- The first term is 17; the ratio is 2.
- From term 1 to term n there are n−1 steps. Multiply by the ratio at each step.
- At n=5, 17·24=272. A recursive rule needs a starting term; an explicit rule calculates any term directly.
Answer: aₙ = 17·2n−1; a5 = 272
Verify: Undo the n−1 changes to recover the first term.
Avoid this mistake: Using n instead of n−1 with indexing starting at 1.
Example 5 · Challenge
Given a₁=17 and aₙ=aₙ₋₁+3 for n≥2, find a6.
| n | aₙ |
|---|---|
| 1 | 17 |
| 2 | 20 |
| 3 | 23 |
| 4 | 26 |
| 5 | 29 |
- The first term is 17; the difference is 3.
- From term 1 to term n there are n−1 steps. Add the common difference at each step.
- At n=6, 17+3·5=32. A recursive rule needs a starting term; an explicit rule calculates any term directly.
Answer: aₙ = 17 + 3(n−1); a6 = 32
Verify: Undo the n−1 changes to recover the first term.
Avoid this mistake: Omitting the starting term of a recursive rule.
Common mistakes and corrections
- Using n instead of n−1 with indexing starting at 1.
- Omitting the starting term of a recursive rule.
- Confusing a common difference with a common ratio.
Real-world applications
- Model rows of seats growing by a fixed count. Define the input and its units, apply the mathematical relationship, and check that the result fits the setting.
- Predict a savings plan or population changing by a fixed factor. Define the input and its units, apply the mathematical relationship, and check that the result fits the setting.
Topic summary
Supply an initial term and apply a repeated process to generate later terms. Compare arithmetic and geometric recursion.
A sequence is an ordered list with a domain of integer indices, usually n≥1. An arithmetic sequence has constant difference d; a geometric sequence has constant ratio r when the required divisions are defined.
Frequently asked questions
What should I know before recursive rules?
Identify repeated differences or ratios and evaluate powers.
How can I check my work?
Undo the n−1 changes to recover the first term. A numerical check can detect an error; when claiming an identity for all inputs, also explain the algebraic property that makes it true.
Which mistakes should I watch for?
Using n instead of n−1 with indexing starting at 1. Omitting the starting term of a recursive rule. Confusing a common difference with a common ratio.
Can I use another valid method?
Yes. Explain your assumptions, show enough intermediate work to justify the result, and verify it in the original problem. Equivalent expressions and valid alternative methods receive credit.
Mathematical process practice · A.1A–A.1G
- A.1A: Model rows of seats growing by a fixed count.
- A.1B: Plan, solve, check, and interpret the written response.
- A.1C: Select paper, a calculator, a table, or a graph; explain the choice.
- A.1D: Connect symbols with a verbal explanation and another representation.
- A.1E: Organize the solution with a table, diagram, or labeled equations.
- A.1F: Explain how the representation and the algebraic rule express the same relationship.
- A.1G: Justify each operation and the reasonableness of the conclusion.
Quick revision notes
- Supply an initial term and apply a repeated process to generate later terms. Compare arithmetic and geometric recursion.
- Using n instead of n−1 with indexing starting at 1.
- Omitting the starting term of a recursive rule.
- Confusing a common difference with a common ratio.
Practice and explain
Try all four multiple-choice questions, then show your reasoning in the written response. Objective scores and AI feedback assess different skills.
Continue learning
Original Cambridge Veritas practice. These are not official STAAR items. Cambridge Veritas is not affiliated with or endorsed by TEA.
Official Algebra I TEKS (§111.39) · TEA assessed curriculum · STAAR blueprint · TEA instructional materials