Algebra I · 35 minutes
Suggested grades 9–10 · Unit 3: Quadratics and discrete patterns · Lesson 2 of 2
Arithmetic and Geometric Sequences
Translate recursive patterns into explicit formulas with a stated first index.
Mr. S Teaches · Original starter lesson · HS-ALG1-06 · Grade range is planning guidance.Watch, notice, discuss
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Introduction to arithmetic sequences | Sequences, series and induction | Precalculus | Khan Academy
By Khan Academy
Build a constant-difference rule and compare recursive and explicit descriptions.
Open on YouTube: Introduction to arithmetic sequences | Sequences, series and induction | Precalculus | Khan Academy (opens a new tab) ↗Geometric sequences | Sequences, series and induction | Precalculus | Khan Academy
By Khan Academy
Contrast constant ratios with the arithmetic pattern and predict a later term.
Open on YouTube: Geometric sequences | Sequences, series and induction | Precalculus | Khan Academy (opens a new tab) ↗How can you tell whether the pattern has a constant difference or a constant ratio?
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Learning objective
Write recursive and explicit rules for two sequences and evaluate a later term.
Before you begin
- Function notation
- Integer exponents
Gather your materials
- Paper and pencil
- Graph paper
- Calculator for checking results
Language to know
- arithmetic sequence
- A sequence with a constant difference between successive terms.
- geometric sequence
- A sequence with a constant ratio between successive nonzero terms.
- recursive rule
- A rule using earlier terms and an initial value.
Understand the concept
A sequence is a function on selected integer inputs. Stating whether counting begins at 0 or 1 prevents off-by-one errors. Arithmetic sequences add a difference; geometric sequences multiply by a ratio. An explicit formula reaches a selected term directly, while a recursive rule describes how successive terms arise.
Teach it step by step
- Index 7,11,15,19 as terms a₁ through a₄ and identify difference 4.
- Write a₁=7, aₙ=aₙ₋₁+4 and derive aₙ=7+4(n−1).
- Index 3,6,12,24 as b₁ through b₄; write ratio-2 recursive and explicit rules.
- Evaluate the sixth term of each and test the explicit formulas at n=1.
Make the thinking visible
Worked example or model response
a₆=7+4(5)=27. For b₁=3 and bₙ=2bₙ₋₁, the explicit rule is bₙ=3·2ⁿ⁻¹, so b₆=96. Using exponent n would incorrectly give b₁=6.
Guided practice
Write both rule forms for 10,7,4,1 and 5,15,45,135, with first index 1.
Independent practice
1. Find the eighth term of 2,7,12,… . 2. Find the fifth term of 80,40,20,… . 3. State each recursive starting value.
Quick assessment
Why is a recursive rule incomplete without an initial term?
Answer key & assessment criteria
Answers and teaching guidance
- Guided: a₁=10, aₙ=aₙ₋₁−3, aₙ=10−3(n−1). b₁=5, bₙ=3bₙ₋₁, bₙ=5·3ⁿ⁻¹.
- Independent 1: 37; a₁=2 and difference 5. 2: 5; b₁=80 and ratio 1/2.
- Quick check: The recurrence describes change but cannot identify which sequence to generate without a start.
Look for: Require the learner to explain the interpretation and check, not only state a result. Distinguish an exact value from a rounded approximation and attend to the stated domain.
Make it work for your learner
More support
Use an indexed term table and annotate how many changes occur before a given term.
Extension challenge
Model a shrinking geometric pattern with a ratio between zero and one and explain its contextual domain.
Verified references · Selected contributions
Standards connections
Subject-content standards
These records identify the part of a subject standard practiced here. Course placement is instructional guidance; the source’s category or band remains authoritative.
Contributes to Common Core standard
HSF.BF.A.2
Common Core · High school · Functions
Regular high school mathematics standard
Write and translate arithmetic and geometric recursive and explicit rules.
Read the official standard ↗This lesson contributes selected skills; broader applications and full standard coverage require further instruction.
Verified 2026-11-04. Selected contribution, not full attainment.