HSF.BF.A.2: Arithmetic and Geometric Sequences, Recursive and Explicit
In plain English: HSF.BF.A.2 is the Common Core functions standard that asks students to write arithmetic and geometric sequences both recursively and with an explicit formula, use them to model situations, and translate between the two forms. Arithmetic sequences add a common difference and geometric sequences multiply by a common ratio. It is usually taught in Algebra I and revisited in Algebra II.
Write arithmetic and geometric sequences both recursively and with an explicit formula, use them to model situations, and translate between the two forms.
Common Core State Standards for Mathematics · Domain: Building Functions (BF) · Cluster: Build a function that models a relationship between two quantities Also written as HSF-BF.A.2 or F-BF.2 · Official standard
Students learn to describe a sequence in two ways. A recursive formula gives the first term and a rule for getting each term from the one before, such as a1 = 7, an = an-1 + 4. An explicit formula gives any term directly from its position n, such as an = 4n + 3. The lesson treats both arithmetic sequences (a common difference is added) and geometric sequences (a common ratio multiplies), and students practice translating in both directions.
Students then use sequences to model situations such as a tree that grows the same amount each year, a ball whose rebounds shrink by a constant percent, and salary offers that grow by a fixed amount or a fixed percent. Throughout, students pay attention to where the count starts (n = 1 or n = 0), because that choice changes the explicit formula.
Learning Objectives
By the end of this lesson, students will be able to:
Write a recursive formula for an arithmetic or a geometric sequence from its terms or from a context
Write an explicit formula for an arithmetic or a geometric sequence and use it to find any term
Translate a recursive formula into an explicit formula and an explicit formula into a recursive formula, including when the count starts at n = 0
Model a situation with an arithmetic or geometric sequence, choose the type from the context, and interpret terms in context
Prior Knowledge Required
Students should already be comfortable with:
Sequences as functions whose domain is a subset of the integers HSF.IF.A.3
Linear functions and constant rate of change 8.F.B.4
Integer exponents and evaluating powers 8.EE.A.1
Percent increase and decrease as multipliers, such as 1.05 and 0.88
Draw the first three figures of a toothpick pattern: a row of 1 square, then 2 squares, then 3 squares sharing sides. Next to it, fold a sheet of paper in half three times in front of the class and count the layers.
Warm-Up Prompt
"Figure 1 uses 4 toothpicks, figure 2 uses 7 and figure 3 uses 10. How many toothpicks does figure 10 use? Figure 100? One fold gives 2 layers, two folds give 4 and three folds give 8. How many layers would 10 folds give? What is different about how the two patterns grow?"
Collect answers: figure 10 uses 31 toothpicks and figure 100 uses 301, because each new square adds 3. Ten folds would give 1,024 layers, because each fold doubles the layers. Students usually describe both patterns step by step first ("add 3", "double it"). Name these as recursive descriptions, then ask how they found figure 100 without counting 99 steps: that is the explicit formula.
Direct Instruction25 minutes
Part 1: The four formulas. Use Diagram 2 to set up the vocabulary. For an arithmetic sequence with first term a1 and common difference d, and a geometric sequence with first term a1 and common ratio r:
Arithmetic, recursive: a1 given, an = an-1 + d for n ≥ 2.
Arithmetic, explicit: an = a1 + (n - 1)d. There are n - 1 steps from the first term to term n.
Geometric, recursive: a1 given, an = r · an-1 for n ≥ 2.
Geometric, explicit: an = a1 · rn-1.
Check any formula by computing the first three terms from it and comparing them with the sequence.
Stress that a recursive formula is not complete without its first term: an = an-1 + 4 describes every sequence that goes up by 4. Then work through the examples below, asking each time: "Is the change added or multiplied?"
Arithmetic: recursive to explicit
The sequence 7, 11, 15, 19, ... has a1 = 7 and an = an-1 + 4. Write the explicit formula and find a20.
Equation: an = 7 + 4(n - 1) = 4n + 3, so a20 = 83
Geometric: terms to both forms
The sequence 5, 15, 45, 135, ... has common ratio 15/5 = 3. Write both formulas and find a8.
Equation: a1 = 5, an = 3 · an-1; an = 5 · 3n-1, so a8 = 10,935
Arithmetic model
A newly planted tree is 1.2 m tall at the end of year 1 and grows 0.45 m each year. Write a formula for its height at the end of year n and find the height at the end of year 10.
Equation: an = 1.2 + 0.45(n - 1), so a10 = 5.25 m
Geometric model
A ball's first rebound reaches 150 cm, and each rebound reaches 60% of the height of the one before. Write a formula for the height of rebound n and find the fourth rebound.
Equation: an = 150(0.6)n-1, so a4 = 32.4 cm
Explicit to recursive, two starting points
Write recursive formulas for an = 50 - 6n (n ≥ 1) and for an = 3 · 2n (n ≥ 1).
Equation: a1 = 44, an = an-1 - 6; and a1 = 6, an = 2 · an-1
For the last example, point out that the first term is found by substituting n = 1, not by reading the number in front: 50 - 6(1) = 44 and 3 · 2¹ = 6. The number being added (-6) or the ratio (2) is read from the formula. Use Diagram 1 to compare the two types: arithmetic terms lie on a line because the steps are equal, while geometric terms with r > 1 take steps that keep growing, so they pass the arithmetic sequence at n = 7.
Guided Practice15-20 minutes
Pairs work through four problems, one at a time. After each, one pair shows its work on the board:
Write both forms for 30, 26, 22, 18, ... (a1 = 30, an = an-1 - 4; an = 34 - 4n.)
Write both forms for 2, 8, 32, 128, ... (a1 = 2, an = 4 · an-1; an = 2 · 4n-1.)
A theater's first row has 16 seats and each row has 3 more than the row in front. Write a formula for row n and find row 12. (an = 13 + 3n; 49 seats.)
A lake's algae covers 40 m² in week 1 and the area grows 30% each week. Write both forms and find the area in week 4. (a1 = 40, an = 1.3 · an-1; an = 40(1.3)n-1; 87.88 m².)
Listen for these errors: writing an = 30 - 4n (the first term used as the constant without adjusting), using the percent 0.3 as the ratio instead of 1.3, and leaving out the first term in a recursive formula.
Independent Practice15 minutes
Students work alone on four problems and check each explicit formula by computing a1 and a2 from it:
Explicit formula for -12, -5, 2, 9, ... (an = 7n - 19.)
Recursive formula for an = 1000(0.9)n-1. (a1 = 1000, an = 0.9 · an-1.)
A job pays $600 in week 1 and $15 more each week after. What is the pay in week 20? (an = 585 + 15n; $885.)
Explicit formula for a0 = 3, an = 2 · an-1 with the count starting at n = 0. (an = 3 · 2n.)
Closure5-10 minutes
Exit ticket: (1) Write the explicit formula for a1 = 9, an = an-1 + 5. (an = 5n + 4.) (2) Write the recursive formula for an = 7 · 2n-1. (a1 = 7, an = 2 · an-1.) (3) A phone loses 20% of its value each year. Is its value a year-by-year arithmetic or geometric sequence? Explain in one sentence. (Geometric: each year's value is 0.8 times the value before.)
Differentiation Strategies
For Struggling Students
Give a table with columns n, term, and "how I got it" so students write the recursive step next to each term before looking for a formula
Color-code the first term and the common difference or ratio in every formula
Start every explicit formula from the template a₁ + (n - 1)d or a₁ · rⁿ⁻¹ before simplifying
For Advanced Students
Find the first term and common difference of an arithmetic sequence given only a₄ = 19 and a₉ = 44
Show that the terms of a geometric sequence with r > 0 have equal ratios, so their logarithms form an arithmetic sequence (a preview of later work)
Write a recursive formula for a sequence that is neither arithmetic nor geometric, such as aₙ = 2aₙ₋₁ + 1 with a₁ = 1, and look for an explicit formula
Assessment Guidance
What to Look For
Check that every recursive formula includes a first term, and that students can say whether a context adds a constant or multiplies by a constant before writing anything. For explicit formulas, check the starting index: a student who writes aₙ = 7 + 4n for 7, 11, 15, ... has started the count at n = 0 without saying so. For percent contexts, look for the ratio written as 1 + rate or 1 - rate, such as 1.3 or 0.6, not as the rate itself.
02
Classroom Activities
3 Activities
1
Four-Way Sequence Match
20 minGroups of 3-4
Each group gets a shuffled set of 16 cards: 4 sequences, each shown as a context, a list of terms, a recursive formula and an explicit formula. Groups sort the cards into 4 matching sets of 4 and justify every match.
Card Sets (with answers)
Set A. Context: a movie club costs $3 to join and $9 per movie; total cost after n movies. Terms: 12, 21, 30, 39. Recursive: a₁ = 12, aₙ = aₙ₋₁ + 9. Explicit: aₙ = 9n + 3
Set B. Context: 3 people receive a message in round 1, and each person who receives it sends it to 2 new people in the next round. Terms: 3, 6, 12, 24. Recursive: a₁ = 3, aₙ = 2 · aₙ₋₁. Explicit: aₙ = 3 · 2ⁿ⁻¹
Set C. Context: a tournament starts with 96 players, and half are eliminated each round; players in round n. Terms: 96, 48, 24, 12. Recursive: a₁ = 96, aₙ = 0.5 · aₙ₋₁. Explicit: aₙ = 96(0.5)ⁿ⁻¹
Set D. Context: a car has 40 liters of fuel at the first stop and uses 5 liters between stops. Terms: 40, 35, 30, 25. Recursive: a₁ = 40, aₙ = aₙ₋₁ - 5. Explicit: aₙ = 45 - 5n
Procedure
Groups first sort the cards into arithmetic and geometric piles, then into the 4 sets
For each set, one student checks that the explicit formula gives the first three terms
Each group explains one match that the members disagreed about
Discussion Questions
In Set C, how many rounds does the tournament last? Why does the sequence stop?
Why is Set A's explicit formula 9n + 3 and not 9n + 12?
Challenge Variation
Groups write a fifth set of 4 cards of their own, trade with another group, and sort the new set.
2
Which Job Offer?
25 minPairs
Pairs compare two invented job offers with a spreadsheet or graphing tool. Job A pays $42,000 in year 1 with a $1,500 raise each year. Job B pays $40,000 in year 1 with a 5% raise each year. Students model each offer with a sequence in both forms and decide which is better over different time spans.
Procedure
Write both forms for each job. Job A: a₁ = 42000, aₙ = aₙ₋₁ + 1500; aₙ = 42000 + 1500(n - 1). Job B: b₁ = 40000, bₙ = 1.05 · bₙ₋₁; bₙ = 40000(1.05)ⁿ⁻¹
Build a spreadsheet with one column per job, using the recursive rule (each cell refers to the cell above), and a second pair of columns using the explicit rule; the columns must match
Find the salaries in year 5 (Job A: $48,000; Job B: $48,620.25) and year 10 (Job A: $55,500; Job B: $62,053.13)
Find the first year in which Job B pays more (year 5: in year 4 Job A pays $46,500 and Job B pays $46,305)
Discussion Questions
Why does a fixed-percent raise eventually beat a fixed-dollar raise?
Which job would you take if you planned to stay 3 years? 10 years?
Which form was easier for the spreadsheet, and which for finding year 30 directly?
Modification for Distance Learning
Share a spreadsheet template with the headings filled in. Pairs work in the same file and add a line chart of both columns.
3
Translation Relay
20 minGroups of 3
Groups pass one sheet around. Each student translates one formula into the other form, and the next student checks it by computing the first three terms from both forms before doing the next translation.
In prompts 5 and 6, how did the starting index change the formula?
How can you read the common difference directly from an explicit arithmetic formula?
03
Diagrams & Visual Aids
2 diagrams
Diagram 1: Arithmetic and Geometric Terms Compared
The first 8 terms of the arithmetic sequence a₁ = 2, aₙ = aₙ₋₁ + 3 (circles) and the geometric sequence a₁ = 2, aₙ = 1.5 · aₙ₋₁ (squares), drawn to scale. The terms are separate points because a sequence is only defined at whole numbers n. The geometric terms first pass the arithmetic terms at n = 7 (about 22.78 versus 20).
Diagram 2: One Step at a Time or a Jump to Term n
The sequences 7, 11, 15, ... and 5, 15, 45, ... from the worked examples. The recursive rule describes each arrow; the explicit rule counts n - 1 arrows from the first term at once.
04
Homework Assignment
~30 min
HSF.BF.A.2 Homework: Arithmetic and Geometric Sequences
Directions: Show your work. For every formula you write, state whether the sequence is arithmetic or geometric and check your formula by computing the first three terms from it. For context problems, say what n and the terms represent, with units.
Part 1: Writing Both Forms (Problems 1-2)
For the sequence -8, -3, 2, 7, ..., write a recursive formula and an explicit formula, then find a25.
For the sequence 1024, 256, 64, 16, ..., write a recursive formula and an explicit formula, then find a7.
Part 2: Translating Between Forms (Problems 3-4)
(a) Write an explicit formula for a1 = 14, an = an-1 + 2.5. (b) Write a recursive formula for an = 81(2/3)n-1.
Find and fix the error in each translation. (a) A student says the recursive form of an = 3n + 10 is a1 = 10, an = an-1 + 10. (b) A student says the recursive form of an = 5 · 2n (n ≥ 1) is a1 = 5, an = 2 · an-1.
Part 3: Modeling (Problems 5-6)
A new streaming service has 2,400 subscribers in month 1 and gains 350 subscribers each month. Write both forms of a sequence for the number of subscribers in month n. In which month does it first have at least 10,000 subscribers?
A town has 18,000 residents in year 1, and its population grows 2% each year. Write both forms of a sequence for the population in year n, find the population in year 6 to the nearest person, and explain why an arithmetic sequence would not fit this description.
Rubric
Criterion
Full Credit (2 pts)
Partial Credit (1 pt)
No Credit (0 pts)
Recursive Formulas
First term and rule both correct
Rule correct, first term missing or wrong
Missing or incorrect
Explicit Formulas
Correct formula, checked with the first three terms
Correct type, index or constant off by one step
Missing or incorrect
Translating Forms
Both directions correct; errors found and explained
One direction correct
Missing or incorrect
Modeling
Correct type chosen and justified; answers interpreted in context
Correct values without interpretation
Wrong type of sequence
05
Quiz: 20 Questions
Interactive, with answers
Instructions
Work through the questions in order. Your score updates as you answer, and Reset quiz clears everything so you or your students can try again. Unless a question says otherwise, sequences start at n = 1.
Multiple choice: pick an option to check it. Short answer: write your answer, then reveal the model answer.
0 of 20 answered · 0 correct
Question 1 of 20 · Multiple Choice
Which recursive formula describes 11, 8, 5, 2, ...?
Answer: B
Each term is 3 less than the one before, and the first term is 11. Choice A adds instead of subtracting. Choice C uses the fourth term as the first. Choice D has no first term, so it does not define one sequence.
Question 2 of 20 · Multiple Choice
Which explicit formula describes 6, 13, 20, 27, ...?
Answer: C
The common difference is 7, so an = 6 + 7(n - 1) = 7n - 1. Check: n = 1 gives 6. Choice D uses 6 + 7n, which gives 13 for n = 1: the count has started at 0. Choice B expands 7(n - 1) as 7n - 1 instead of 7n - 7. Choice A swaps the first term and the difference.
Question 3 of 20 · Multiple Choice
Which explicit formula describes 4, 12, 36, 108, ...?
Answer: A
The ratio is 12/4 = 3 and the first term is 4, so the explicit formula is 4 · 3 to the power n - 1. Choice C gives 12 for n = 1. Choice D treats the sequence as arithmetic: it gives 4, 12, 20, which does not match the third term.
Question 4 of 20 · Multiple Choice
Which recursive formula describes 200, 100, 50, 25, ...?
Answer: D
Each term is half of the one before: the common ratio is 0.5, and the first term is 200. Choice A subtracts 100 each time, which works for the first step only (100 - 100 = 0, not 50). Choice B doubles instead of halving.
Question 5 of 20 · Multiple Choice
An arithmetic sequence has an = 5 + 4(n - 1). What is a30?
Answer: B
5 + 4(29) = 5 + 116 = 121. Choice A uses 4(30) instead of 4(29), forgetting that there are only 29 steps from the first term to the 30th. Choice D multiplies 5 by 31, ignoring the order of operations.
Question 6 of 20 · Multiple Choice
A geometric sequence has an = 3 · 2n-1. What is a6?
Answer: A
3 · 2⁵ = 3 · 32 = 96. Choice B uses the exponent 6 instead of 5. Choice C multiplies 3 · 2 · 6. Choice D computes (3 · 2)⁶, raising the whole product to the power.
Question 7 of 20 · Multiple Choice
Which explicit formula matches a1 = 10, an = an-1 + 6?
Answer: D
an = 10 + 6(n - 1) = 6n + 4. Check: n = 1 gives 10 and n = 2 gives 16. Choice B gives 16 for n = 1: the count has started at 0. Choice C expands 6(n - 1) as 6n - 1 instead of 6n - 6. Choice A swaps the first term and the difference.
Question 8 of 20 · Multiple Choice
Which recursive formula matches an = 2 · 5n-1?
Answer: C
Substituting n = 1 gives the first term 2, and the base 5 is the common ratio. Choice A swaps the first term and the ratio. Choice B treats the sequence as arithmetic. Choice D uses 2 · 5 = 10 as the first term, which is the second term.
Question 9 of 20 · Multiple Choice
The explicit formula an = 6 · 3n is used for n ≥ 1. Which recursive formula gives the same sequence?
Answer: B
The first term is 6 · 3¹ = 18, and each term is 3 times the one before. Choice A reads 6 as the first term, but 6 would be the term for n = 0. Choice C uses the coefficient as the ratio.
Question 10 of 20 · Multiple Choice
A gardener plants 8 tulips on day 1 and 5 more tulips each day after that. Which formula gives the total number planted by the end of day n?
Answer: A
This is arithmetic with a1 = 8 and d = 5: an = 8 + 5(n - 1) = 5n + 3. Check: day 2 gives 13. Choice D gives 13 on day 1. Choice C treats a constant addition as multiplication.
Question 11 of 20 · Multiple Choice
A car worth $22,000 in year 1 loses 12% of its value each year. Which formula gives its value in year n?
Answer: D
Losing 12% leaves 88% each year, so the common ratio is 0.88. Choice A keeps only 12% each year. Choice C increases the value by 12%. Choice B subtracts 12 cents a year instead of 12%.
Question 12 of 20 · Multiple Choice
Which situation is modeled by a geometric sequence?
Answer: C
Tripling means multiplying by the same number, 3, each hour: a common ratio. The other three situations add the same amount each time, so they are arithmetic.
Question 13 of 20 · Multiple Choice
A stack of logs has 22 logs in the bottom layer (layer 1), and each layer above it has 2 fewer logs. How many logs are in layer 8?
Answer: C
an = 22 - 2(n - 1), so a8 = 22 - 14 = 8. Choice A uses 2(8) instead of 2(7): there are only 7 steps from layer 1 to layer 8. Choice B adds 2 each layer instead of subtracting. Choice D doubles 22, as if the sequence were geometric.
Question 14 of 20 · Multiple Choice
Which recursive formula matches an = 40 - 3n for n ≥ 1?
Answer: B
a1 = 40 - 3 = 37, and each term is 3 less than the one before. Choice A reads 40 as the first term, but 40 would be the term for n = 0. Choice C adds 3 instead of subtracting.
Question 15 of 20 · Short Answer
Write a recursive formula and an explicit formula for 2.5, 4, 5.5, 7, ... and find the 40th term.
Arithmetic with d = 1.5. Recursive: a1 = 2.5, an = an-1 + 1.5.Explicit: an = 2.5 + 1.5(n - 1) = 1.5n + 1. a40 = 1.5(40) + 1 = 61.
Question 16 of 20 · Short Answer
Write a recursive formula and an explicit formula for 5, -10, 20, -40, ... and find the 9th term.
Geometric with ratio -10/5 = -2. Recursive: a1 = 5, an = -2 · an-1.Explicit: an = 5(-2)n-1. a9 = 5(-2)⁸ = 5 · 256 = 1,280. A negative ratio makes the signs alternate.
Question 17 of 20 · Short Answer
A patient has 400 mg of a medicine in the body at hour 1, and each hour 20% of the amount present leaves the body. Write an explicit formula for the amount at hour n and find the amount at hour 5.
Each hour 80% remains, so the ratio is 0.8: an = 400(0.8)n-1. a5 = 400(0.8)⁴ = 400(0.4096) = 163.84 mg.
Question 18 of 20 · Short Answer
A bike share charges $3.50 to unlock a bike plus $0.25 per minute. Let an be the cost of an n-minute ride. Write both forms and find the cost of a 30-minute ride.
Explain why an = an-1 + 4 alone does not define a sequence. Then write the explicit formula for the sequence with a1 = -3.
The rule only says each term is 4 more than the one before; without a first term, -3, 1, 5, ... and 10, 14, 18, ... both follow it. With a1 = -3: an = -3 + 4(n - 1) = 4n - 7.
Question 20 of 20 · Short Answer
Channel A has 1,000 subscribers in month 1 and gains 200 each month. Channel B has 100 subscribers in month 1 and grows 50% each month (invented numbers). Write an explicit formula for each and find the first month in which B has more subscribers than A.
A: an = 1000 + 200(n - 1). B: bn = 100(1.5)n-1. Month 9: A has 2,600 and B has about 2,563. Month 10: A has 2,800 and B has about 3,844. B first has more in month 10. Growth by a constant ratio eventually beats growth by a constant difference.
0 of 20 answered · 0 correct
06
Frequently Asked Questions
10 Questions
What does HSF.BF.A.2 mean?
HSF.BF.A.2 asks students to write arithmetic and geometric sequences in two forms, recursive and explicit, to use them to model situations, and to translate from one form to the other. For example, 7, 11, 15, ... can be written as a₁ = 7, aₙ = aₙ₋₁ + 4 or as aₙ = 4n + 3.
Is HSF.BF.A.2 Algebra 1 or Algebra 2?
It is usually taught in Algebra I, often next to linear and exponential functions, and revisited in Algebra II, where students also work with series. The standard itself covers only writing, modeling and translating sequences, not sums.
What is the difference between a recursive and an explicit formula?
A recursive formula gives the first term and a rule to get each term from the previous one, so you build the sequence step by step. An explicit formula gives term n directly from n. To find the 100th term, the explicit formula is faster; to describe a process like "add 4 each week", the recursive formula often matches the wording better.
How do you tell if a sequence is arithmetic or geometric?
Subtract and divide consecutive terms. If the differences are all equal, the sequence is arithmetic; if the ratios are all equal, it is geometric. In a context, "the same amount more" suggests arithmetic and "the same percent" or "doubles" suggests geometric. Some sequences are neither, such as 1, 4, 9, 16.
Why is it n - 1 in the explicit formula?
Because term n is n - 1 steps after the first term. In aₙ = a₁ + (n - 1)d, the first term needs 0 steps of d, the second needs 1, and so on. If a context starts counting at n = 0 (for example, the starting amount before any years pass), the formula becomes aₙ = a₀ + nd or aₙ = a₀ · rⁿ.
What is a common mistake when translating an explicit formula to a recursive one?
A common one is reading the first term from the constant in the formula. For aₙ = 50 - 6n, the first term is 50 - 6 = 44, not 50. For aₙ = 3 · 2ⁿ, the first term is 6, not 3. Students should always substitute n = 1 (or the stated starting index) to find the first term.
How are sequences connected to linear and exponential functions?
An arithmetic sequence is a linear function restricted to whole-number inputs: the common difference is the slope. A geometric sequence is an exponential function restricted to whole numbers: the common ratio is the growth factor. This is why Common Core places HSF.LE.A.2, constructing linear and exponential functions including sequences, next to this standard.
How is HSF.BF.A.2 tested?
Typical items ask students to choose or write a formula for a list of terms, to translate a recursive formula into an explicit one or the reverse, to find a specific term, or to write a sequence for a short context such as a salary with yearly raises or a population that grows by a percent. Some items give a recursive rule and ask for a later term, which students can find with either form.
Can a geometric sequence have a negative or fractional ratio?
Yes. With r = 0.5, the terms halve each time, as in a tournament where half the players are eliminated each round. With r = -3, the signs alternate: 2, -6, 18, -54. In modeling contexts the ratio is usually positive, such as 0.85 for a 15% loss or 1.05 for a 5% gain.
How can students check a sequence formula?
Compute the first three terms from the formula and compare them with the given terms or the context. This quickly catches an index that is off by one or a percent used as the ratio. A spreadsheet with one column for each form is a quick way to check many terms at once.
07
Related Standards
6 standards
These standards connect to HSF.BF.A.2: prerequisites to review first, parallel standards at the same level, and next steps that build on it.
Before this lesson
HSF.IF.A.3Prerequisite
Recognize that sequences are functions, sometimes defined recursively, on the integers
