HSF.IF.A.3: Sequences as Functions and Recursive Definitions
In plain English: HSF.IF.A.3 is the Common Core functions standard that asks students to recognize that a sequence is a function whose domain is a subset of the integers, such as {1, 2, 3, ...}, and that sequences are sometimes defined recursively, with each term built from earlier terms. Its official example is the Fibonacci sequence. It is usually taught in Algebra I.
Recognize that sequences are functions, sometimes defined recursively, whose domain is a subset of the integers. For example, the Fibonacci sequence is defined recursively by f(0) = f(1) = 1, f(n+1) = f(n) + f(n-1) for n ≥ 1.
Common Core State Standards for Mathematics · Domain: Interpreting Functions (IF) · Cluster: Understand the concept of a function and use function notation Also written as HSF-IF.A.3 or F-IF.3 · Official standard
Students connect sequences to the function ideas they already have. A sequence such as 7, 11, 15, 19, ... is a function: the input is the term number n and the output is the term. What makes it special is the domain, which is a set of integers such as {1, 2, 3, ...}, {0, 1, 2, ...} or a finite set like {1, 2, ..., 20}. That is why the graph of a sequence is a set of separate points.
The second idea is that a sequence can be defined recursively: a starting value, or several, plus a rule that builds each term from earlier ones. Students read recursive definitions, compute terms from them and work through the official example, the Fibonacci sequence, where each term depends on the two before it.
Learning Objectives
By the end of this lesson, students will be able to:
Explain why a sequence is a function and identify its domain as a subset of the integers
Decide whether a situation is naturally described by a sequence, and state the domain
Compute terms from a recursive definition, including rules that use the two previous terms
Work through the Fibonacci definition f(0) = f(1) = 1, f(n + 1) = f(n) + f(n - 1), explaining the role of each starting value
Graph a sequence as separate points and explain why the points are not connected
Prior Knowledge Required
Students should already be comfortable with:
Functions, domain, range and f(x) notation HSF.IF.A.1
Evaluating functions for inputs in their domains HSF.IF.A.2
Describing patterns in numerical sequences 5.OA.B.3
Build a tile pattern on the document camera: Figure 1 is an L shape of 3 squares, Figure 2 adds one square to each arm (5 squares), Figure 3 has 7 squares.
Warm-Up Prompt
"How many squares will Figure 10 have? Could there be a Figure 2.5? What about Figure 0? What are the possible inputs for this pattern?"
Students usually find 21 squares for Figure 10 (each figure adds 2). The question about Figure 2.5 is the point of the warm-up: the input is a figure number, and figure numbers are whole numbers. Record the pattern as a table with columns n and squares, and ask whether each figure number has exactly one number of squares. It does, so the pattern is a function whose inputs are only the integers 1, 2, 3, ...
Direct Instruction20 minutes
Sequences are functions. A sequence assigns to each term number n exactly one term. Write it as f(n), or as an, which means the same as a(n). The domain is a subset of the integers: often {1, 2, 3, ...}, sometimes {0, 1, 2, ...}, and sometimes a finite set. Show Diagram 1: the sequence 2, 5, 8, 11, 14 and the function y = 3x - 1 share a rule, but the sequence is only defined at integer inputs, so its graph is five points.
Recursive definitions. Instead of a formula in n, a sequence can be given by starting value(s) and a rule that says how to get a term from earlier terms. Model how to read one:
Find the starting value or values: these are the terms the rule cannot produce, such as g(1) = 3.
Find where the rule applies: "for n ≥ 2" means the rule gives g(2), g(3) and so on.
Substitute the smallest allowed n: g(2) = 2g(1) = 6.
Repeat, one term at a time: each new term needs the earlier ones, so recursive rules are computed in order.
A sequence written as a function
The sequence 7, 11, 15, 19, ... starts with f(1) = 7. Name the domain, find f(4), and check that f(n) = 4n + 3 gives the same terms.
Equation: Domain {1, 2, 3, ...}; f(4) = 19; 4(1) + 3 = 7 and 4(4) + 3 = 19. f(2.5) is not a term
Reading a recursive definition
g(1) = 3 and g(n) = 2g(n - 1) for n ≥ 2. List the first five terms.
A camp water tank holds 50 liters, and 6 liters are used each day: a(0) = 50 and a(n) = a(n - 1) - 6. The sequence stops when the tank can no longer supply a full day.
Equation: a(8) = 2, so the domain is {0, 1, 2, ..., 8}, a finite set of integers
Spend time on the Fibonacci example, the official example in the standard. Point out that the rule f(n + 1) = f(n) + f(n - 1) uses two earlier terms, so it needs two starting values, and that it applies for n ≥ 1: substituting n = 1 gives f(2) = f(1) + f(0). Use Diagram 2 to show each term as the sum of the two boxes before it, and the graph of the terms as separate points.
Guided Practice15 minutes
Pairs work through four tasks and compare with another pair after each: (1) t(1) = 10 and t(n) = t(n - 1) + 7 for n ≥ 2; list five terms (10, 17, 24, 31, 38). (2) s(0) = 1 and s(n) = 3s(n - 1) for n ≥ 1; list s(0) to s(4) (1, 3, 9, 27, 81) and name the domain. (3) Decide which is a sequence: the number of seats in each row of a 12-row bleacher, or the weight of a puppy over its first year (the first; the second changes continuously). (4) In the Fibonacci definition, what equation does n = 3 produce? (f(4) = f(3) + f(2).) Listen for students who start task (2) at s(1), and for students who think the rule in task (4) can be used for n = 0.
Independent Practice10-15 minutes
Students work alone: (1) the sequence 40, 36, 32, 28, ... starts at f(1) = 40; find f(5) (24) and give the domain; (2) r(1) = 1 and r(n) = 2r(n - 1) + 1 for n ≥ 2; list five terms (1, 3, 7, 15, 31); (3) f(n) = n² with domain {1, 2, 3, 4, 5, 6}; explain whether f(0) is a term (no, 0 is not in the domain); (4) plot the five terms from item (2) as points (n, r(n)) on graph paper and explain why you did not connect them.
Closure5 minutes
Exit ticket: (1) Why is the graph of a sequence a set of separate points? (2) If b(1) = 4 and b(n) = b(n - 1) - 3 for n ≥ 2, what is b(4)? (Answer: -5.) (3) Name the two parts every recursive definition needs.
Differentiation Strategies
For Struggling Students
Give a two-column table with n already filled in (1, 2, 3, ...) so students only fill in the terms, and draw an arrow from each term to the next one it produces
Have students say the recursive rule in words before using it, for example "each term is 7 more than the one before"
Use tiles or grid paper to build the first few figures of a pattern before writing any notation
For Advanced Students
Ask students to prove that in the Fibonacci sequence every third term, f(2), f(5), f(8), ..., is even, using only the recursive rule and the parity of the starting values
Ask students to invent a recursive rule that uses three previous terms, and to decide how many starting values it needs
Ask whether a sequence could have the domain {..., -2, -1, 0, 1, 2, ...}, and to extend the Fibonacci sequence backward to f(-1) and f(-2) using the same rule
Assessment Guidance
What to Look For
Check that students name a domain for every sequence, including where it starts, and that they know the outputs of a sequence need not be integers even though the inputs are. With recursive rules, look for terms computed in order from the stated starting value, and for the correct first n: a common error is applying "for n ≥ 2" to n = 1. For the Fibonacci definition, a strong answer explains that two starting values are needed because each new term uses two earlier ones. Students who connect the points of a sequence graph with a line have not yet linked the graph to the domain.
02
Classroom Activities
3 Activities
1
Sequence or Not? Domain Sort
15 minPairs
Pairs sort 8 situation cards into two groups: those naturally described by a sequence (the inputs are integers) and those described by a function of a continuous input. For each sequence, pairs write the domain as a set.
The 8 Cards (answers for the teacher)
Card 1: the number of seats in row n of a theater with 20 rows: sequence, domain {1, 2, ..., 20}
Card 2: the outside temperature t hours after midnight: not a sequence, because time is continuous
Card 3: your savings balance at the end of month n of one year: sequence, domain {1, 2, ..., 12}
Card 4: the height of a ball t seconds after it is thrown: not a sequence
Card 5: the number of handshakes when n people each shake hands once with everyone else: sequence, domain {2, 3, 4, ...}
Card 6: the area of a square with side length s centimeters: not a sequence
Card 7: the number of cells after n rounds of division, starting from one cell: sequence, domain {0, 1, 2, ...}
Card 8: the cost of n concert tickets at $35 each, with a limit of 8 per customer: sequence, domain {1, 2, ..., 8}
Procedure
Pairs sort the cards and write the domain on each sequence card
For each non-sequence card, pairs describe how measuring only at whole-number times would turn it into a sequence
Two pairs compare domains, especially where they start
Discussion Questions
Why does Card 5 start at 2 and Card 7 start at 0?
Card 2 becomes a sequence if the temperature is read once an hour. What is its domain then?
Can the outputs of a sequence be fractions or negative numbers?
2
Recursive Rule Relay
15 minGroups of 4
Each group gets one recursive rule card. The first student writes the starting value or values, and each student in turn applies the rule once and passes the paper on, until the group has six terms. The relay makes the recursive idea physical: nobody can compute a term before the previous one exists.
Rule Cards (answers for the teacher)
Card A: p(1) = 5 and p(n) = p(n - 1) + 9 for n ≥ 2: 5, 14, 23, 32, 41, 50
Card B: q(0) = 2 and q(n) = 5q(n - 1) for n ≥ 1: 2, 10, 50, 250, 1250, 6250
Card C: w(1) = 64 and w(n) = w(n - 1)/2 for n ≥ 2: 64, 32, 16, 8, 4, 2
Card D: z(1) = 1, z(2) = 2 and z(n) = z(n - 1) · z(n - 2) for n ≥ 3: 1, 2, 2, 4, 8, 32
Procedure
Groups complete their card, then swap cards with another group and repeat
After two rounds, each group plots one of its sequences as points (n, term)
Groups check their terms against another group that did the same card
Discussion Questions
Why does Card D need two starting values while the others need one?
Which card has a domain starting at 0? How does that change the name of the sixth term?
For Card C, what happens to the terms after the sixth one? Are they still terms of the sequence?
3
Fibonacci Investigation
20 minPairs
Pairs extend the official Fibonacci sequence, f(0) = f(1) = 1 and f(n + 1) = f(n) + f(n - 1) for n ≥ 1, and look for patterns that follow from the recursive rule.
Tasks
Extend the sequence from f(0) to f(12). (Terms: 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233.)
Circle the even terms. What pattern do you see in their positions? (f(2), f(5), f(8) and f(11): every third term.)
Add f(0) + f(1) + ... + f(5) and compare the sum with f(7). (The sum is 20, one less than f(7) = 21.) Test the same idea for another n
Explain, using the rule, why the sequence can never have two even terms in a row
Discussion Questions
What would change if the starting values were f(0) = 2 and f(1) = 1 but the rule stayed the same?
Why does the rule say "for n ≥ 1" and not "for n ≥ 0"?
Is f(1.5) a term? Why or why not?
Going Further Variation
Pairs compute the ratios f(n + 1)/f(n) for n = 1 to 11 and describe what happens to them. (They settle near 1.618, the golden ratio.) This goes beyond the standard and is optional.
03
Diagrams & Visual Aids
2 diagrams
Diagram 1: A Sequence Versus a Function of a Real Variable
Drawn to scale. The sequence u(n) = 3n - 1 with domain {1, 2, 3, 4, 5} is a function whose graph is five separate points. The function y = 3x - 1, defined for every real x, uses the same rule, but its graph is an unbroken line (dashed).
Diagram 2: The Fibonacci Sequence Built Recursively
The official example f(0) = f(1) = 1, f(n + 1) = f(n) + f(n - 1) for n ≥ 1. The two starting values are outlined; every later term is the sum of the two terms before it. Below, the points (n, f(n)) for n = 0 to 7, drawn to scale and not connected, because the domain contains only integers.
04
Homework Assignment
~30 min
HSF.IF.A.3 Homework: Sequences as Functions
Directions: Show every term you compute, in order. For each sequence, state its domain as a set, including where it starts.
Part 1: Sequences as Functions (Problems 1-2)
The sequence 9, 14, 19, 24, ... starts with f(1) = 9. (a) State the domain. (b) Find f(6). (c) Explain why f(0) and f(3.5) have no value for this sequence.
Decide whether each situation is naturally a sequence. If it is, give the domain. (a) The height of a sunflower measured every Monday for 10 weeks. (b) The height of the same sunflower at every moment of the summer. (c) The cost c(n) of n movie tickets at $12 each, for 0 to 10 tickets; also find c(7).
Part 2: Recursive Definitions (Problems 3-4)
b(1) = 80 and b(n) = b(n - 1)/2 for n ≥ 2. (a) List the first five terms. (b) Find b(6) and explain why it is still a term of the sequence even though it is not a whole number.
h(0) = 2 and h(n) = 3h(n - 1) - 1 for n ≥ 1. (a) Describe the rule in words. (b) Find h(1), h(2), h(3) and h(4). (c) How many terms must you compute to find h(4), and why?
Part 3: Rules That Use Two Earlier Terms (Problems 5-6)
The Lucas numbers use the Fibonacci rule with different starting values: L(0) = 2, L(1) = 1 and L(n + 1) = L(n) + L(n - 1) for n ≥ 1. (a) List L(0) through L(7). (b) Explain how the terms compare with the Fibonacci terms, and why changing only the starting values changes every later term.
t(1) = 1, t(2) = 3 and t(n) = t(n - 1) + 2t(n - 2) for n ≥ 3. (a) List the first six terms. (b) Is t(0) defined by this definition? Explain. (c) Plot the six terms as points and explain why you do not connect them.
Rubric
Criterion
Full Credit (2 pts)
Partial Credit (1 pt)
No Credit (0 pts)
Domain
Every domain stated as a set with the right starting value
Domains given but one starts at the wrong integer
Domains missing
Recursive Terms
All terms correct and computed in order
One arithmetic error carried forward
Rule applied incorrectly
Two-Term Rules
Both starting values used; terms and comparison correct
Terms correct, explanation incomplete
Only one starting value used
Explanations and Graphs
Clear reasons about inputs being integers; points not connected
Reasons vague
No explanation
05
Quiz: 20 Questions
Interactive, with answers
Instructions
Choose an answer for each question and read the explanation that appears. Short-answer questions show a model answer. Use Reset quiz to clear the page and try again.
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
The sequence 5, 8, 11, 14, ... starts with f(1) = 5. What is its domain?
Answer: A
The inputs of a sequence are the term numbers. Since the first term is f(1), the domain is {1, 2, 3, ...}. Choice B is the range, the set of terms. Choice D would be right only if the first term were called f(0).
Question 2 of 20 · Multiple Choice
Which situation is best described by a sequence?
Answer: B
Day numbers are the integers 1 to 30, so the inputs form a subset of the integers. The other choices use time as a continuous input, which is a function but not a sequence.
Question 3 of 20 · Multiple Choice
g(1) = 4 and g(n) = g(n - 1) + 6 for n ≥ 2. What is g(4)?
Answer: C
Compute in order: g(1) = 4, g(2) = 10, g(3) = 16, g(4) = 22. Choice D is g(3), one step short. Choice A takes one step too many. Choice B multiplies 4 by 6 instead of using the rule.
Question 4 of 20 · Multiple Choice
k(0) = 3 and k(n) = 4k(n - 1) for n ≥ 1. What is k(3)?
Answer: D
k(0) = 3, k(1) = 12, k(2) = 48, k(3) = 192. Choice A is k(2): it starts counting at k(1). Choice C goes one step too far. Choice B multiplies 3 by 4 and then by 3.
Question 5 of 20 · Multiple Choice
Which description fits the graph of a sequence?
Answer: B
A sequence is defined only at integer inputs, so its graph is a set of separate points (n, f(n)). Choices A and C describe a function whose domain is an interval of real numbers. Choice D is not the graph of any function.
Question 6 of 20 · Multiple Choice
Which is a recursive definition of the sequence 6, 11, 16, 21, ...?
Answer: A
Choice A starts at 6 and adds 5 each time: 6, 11, 16, 21. Choice B is not recursive and gives 11, 17, ... Choice C multiplies instead of adding. Choice D swaps the starting value and the difference.
Question 7 of 20 · Multiple Choice
In the official definition of the Fibonacci sequence, f(0) = f(1) = 1 and f(n + 1) = f(n) + f(n - 1) for n ≥ 1. What equation does the rule give for n = 4?
Answer: C
Replace n with 4 everywhere: f(4 + 1) = f(4) + f(4 - 1), that is f(5) = f(4) + f(3). Choice A is what n = 3 gives. Choice B substitutes the term numbers as if they were the terms themselves.
Question 8 of 20 · Multiple Choice
Why does the Fibonacci definition give two starting values, f(0) and f(1)?
Answer: D
The rule f(n + 1) = f(n) + f(n - 1) needs two earlier terms. With only f(0), the first use of the rule (n = 1) would need f(1), which would be unknown. Choice B is false: f(2) = 2 is even.
Question 9 of 20 · Multiple Choice
a(1) = 2 and a(n) = a(n - 1) + n for n ≥ 2. What is a(4)?
Answer: D
The amount added changes with n: a(2) = 2 + 2 = 4, a(3) = 4 + 3 = 7, a(4) = 7 + 4 = 11. Choice A adds 2 at every step. Choice B skips the n = 2 step and computes 2 + 3 + 4. Choice C adds 4 at every step.
Question 10 of 20 · Multiple Choice
A sequence f has the domain {1, 2, 3, 4, 5, 6}. Which statement is true?
Answer: C
A finite domain of six integers means six terms, f(1) through f(6). Choices A and D use inputs outside the domain. Choice B treats the sequence as if its domain included every real number between 1 and 6.
Question 11 of 20 · Multiple Choice
s(1) = 1 and s(n) = s(n - 1) - 3 for n ≥ 2. Which list gives s(1) to s(4)?
Answer: A
Start at 1 and subtract 3 each time. Choice B leaves out the starting value s(1) = 1. Choice C adds 3, and choice D multiplies by -3.
Question 12 of 20 · Multiple Choice
The sequence 2, 4, 8, 16, ... starts with d(1) = 2. Is it a function?
Answer: B
Each input n has exactly one output d(n), so the sequence is a function with domain {1, 2, 3, ...}. How fast the outputs grow does not matter (choice C), and connecting points would add inputs that are not in the domain (choice D).
Question 13 of 20 · Multiple Choice
m(1) = 10, m(2) = 4 and m(n) = m(n - 1) - m(n - 2) for n ≥ 3. What is m(5)?
Answer: A
m(3) = 4 - 10 = -6, m(4) = -6 - 4 = -10, m(5) = -10 - (-6) = -4. Choice D is m(3). Choice B comes from computing -10 - 6 instead of -10 - (-6).
Question 14 of 20 · Multiple Choice
A rule says v(n) = v(n - 1) + 4 for n ≥ 2, but no starting value is given. What can you conclude?
Answer: B
A recursive definition needs a starting value. With v(1) = 0 the terms are 0, 4, 8, ...; with v(1) = 7 they are 7, 11, 15, ... Choices A and C assume a starting value that was never given.
Question 15 of 20 · Short Answer
e(1) = 7 and e(n) = e(n - 1) - 4 for n ≥ 2. List the first five terms and state the domain.
7, 3, -1, -5, -9. The domain is {1, 2, 3, ...}. The terms become negative, which is allowed: only the inputs of a sequence must be integers.
Question 16 of 20 · Short Answer
A sequence is given by f(n) = n² + 1 with domain {0, 1, 2, 3, 4}. List its terms and explain why f(2.5) is not a term.
The terms are 1, 2, 5, 10, 17. The formula can be evaluated at 2.5, but 2.5 is not in the domain of the sequence, so f(2.5) is not one of its terms.
Question 17 of 20 · Short Answer
A sequence uses the Fibonacci rule f(n + 1) = f(n) + f(n - 1) for n ≥ 1, but starts with f(0) = 3 and f(1) = 4. Find f(2) through f(5).
f(2) = 4 + 3 = 7, f(3) = 7 + 4 = 11, f(4) = 11 + 7 = 18, f(5) = 18 + 11 = 29. Same rule as the official Fibonacci sequence, but new starting values give a new sequence.
Question 18 of 20 · Short Answer
c(1) = 1 and c(n) = c(n - 1) + 2n - 1 for n ≥ 2. Find c(2) through c(5) and describe the terms.
c(2) = 1 + 3 = 4, c(3) = 4 + 5 = 9, c(4) = 9 + 7 = 16, c(5) = 16 + 9 = 25. The terms are the perfect squares: c(n) = n², built by adding consecutive odd numbers.
Question 19 of 20 · Short Answer
The rows of a small theater follow r(1) = 18 and r(n) = r(n - 1) + 2 for 2 ≤ n ≤ 15, where r(n) is the number of seats in row n. What is the domain, what does r(4) mean, and what is its value? Why is r(16) not defined?
The domain is {1, 2, ..., 15}. r(4) is the number of seats in row 4: 18, 20, 22, 24 seats. r(16) is not defined because the theater has only 15 rows, so 16 is not in the domain.
Question 20 of 20 · Short Answer
A student graphs the sequence a(n) = 2n with domain {1, 2, 3, 4} and joins the points with a straight line. What is wrong with the graph?
The line includes points such as (1.5, 3) whose inputs are not in the domain. The graph of this sequence is only the four points (1, 2), (2, 4), (3, 6) and (4, 8). The line is the graph of the function y = 2x, not of the sequence.
0 of 20 answered · 0 correct
06
Frequently Asked Questions
10 Questions
What does HSF.IF.A.3 mean?
It means students recognize two things about sequences. First, a sequence is a function: the input is the term number and the output is the term, and the domain is a set of integers such as {1, 2, 3, ...}. Second, a sequence can be defined recursively, by starting values and a rule that builds each term from earlier terms, as in the Fibonacci sequence.
Is HSF.IF.A.3 taught in Algebra 1 or Algebra 2?
It is usually taught in Algebra I, often in the same unit as arithmetic and geometric sequences or right after the definition of a function. Algebra II returns to sequences with series and with the explicit and recursive formulas of HSF.BF.A.2 in more depth.
Why is a sequence a function?
Because each term number has exactly one term. In 7, 11, 15, 19, ..., the input 3 always gives 15 and nothing else, which is the definition of a function (HSF.IF.A.1). The only difference from functions like f(x) = 4x + 3 is the domain, which contains only integers.
What is a recursive definition of a sequence?
A definition that gives one or more starting values and a rule for finding each term from the terms before it. For example, g(1) = 3 and g(n) = 2g(n - 1) for n ≥ 2 give 3, 6, 12, 24, ... A recursive rule by itself is not enough: without the starting value, the terms are not determined.
What is the difference between a recursive and an explicit formula?
A recursive formula finds a term from earlier terms, so you compute the terms in order. An explicit formula finds any term directly from n, such as f(n) = 4n + 3. HSF.IF.A.3 asks students to recognize recursive definitions; writing both kinds of formulas for arithmetic and geometric sequences is the focus of HSF.BF.A.2.
Why does the Fibonacci definition need two starting values?
Its rule, f(n + 1) = f(n) + f(n - 1), adds the two previous terms. The first time the rule is used (n = 1) it needs both f(1) and f(0), so both must be given. A rule that uses only the previous term, such as adding 5, needs one starting value.
Should a sequence start at n = 0 or n = 1?
Either is correct, as long as the definition says which. The official Fibonacci example starts at f(0), while many textbooks start arithmetic sequences at a1. The starting index changes the name of each term: in a sequence that starts at f(0), the fifth term is f(4). Asking students to state the domain every time prevents off-by-one errors.
What is the difference between an and f(n)?
There is none in meaning: both name the term with term number n. Subscript notation an is traditional for sequences, and f(n) is function notation. The standard uses f(n) on purpose, to show that a sequence is a function.
Why shouldn't students connect the points of a sequence graph?
Because the domain has only integers. A line through the points would include inputs such as 2.5, which are not term numbers. Connecting the points turns the graph of the sequence into the graph of a different function with a real-number domain.
What mistakes do students make with recursive sequences?
Common ones are applying the rule to the starting term, for example using "for n ≥ 2" at n = 1; losing count so that the answer is one term off; forgetting a second starting value for a rule that uses two earlier terms; and sign errors when the rule subtracts a negative term. Writing n next to every term, in a table, prevents most of them.
07
Related Standards
6 standards
These standards connect to HSF.IF.A.3: prerequisites to review first, parallel standards at the same level, and next steps that build on it.
Before this lesson
8.F.A.1Prerequisite
Understand a function as a rule assigning each input exactly one output
