Standard: F-IF.3 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.
Clarification: A sequence is a function whose
domain is a subset of the integers. In fact,
many patterns explored in grades k-8 can be
considered sequences. For example, the
sequence 4,7,10, 13, 16… might be described as
a “plus 3 pattern” because terms are computed
by adding 3 to the previous term. To show how
the sequence can be considered a function we
need index that indicates which term of the
sequence we are talking about and which
serves as an input value to the function.
Deciding that the 4 correspondence, as in the
margin. The sequence can be described
recursively by the rule f (1) = 4, f (n+1) = f(n) +3
for n≥2. Notice that the recursive definition
requires both a starting value and a rule for
computing subsequent terms. The sequence
can also be described with the closed formula
f(n)=3n+1, for integers ≥ 1. Notice that the
domain is included as part of the description.
Understandings and notes about standard:
 I can learn the use of an index to indicate which terms
are being used.
 I can use formal notation and language for functions.
 I can understand domain and range as input/output
values.
 I can describe sequences as recursive or closed
formulas.
 I can demonstrate understanding that a graph of the
sequence consists of discrete dots, because the
specification does not indicate what happens “between
the dots.”

Sample Tasks:
Module 3 Lesson 1, Ex 2
Consider the sequence that follows a “plus 3” pattern: 𝟒, 𝟕, 𝟏𝟎, 𝟏𝟑, 𝟏𝟔,….
a.
Write a formula for the sequence using both the 𝒂𝒏 notation and the 𝒇(𝒏) notation.
𝒂𝒏 = 𝟑𝒏 + 𝟏 𝒐𝒓 𝒇(𝒏) = 𝟑𝒏 + 𝟏 starting with 𝒏 = 𝟏.
b.
Does the formula 𝒇(𝒏) = 𝟑(𝒏 − 𝟏) + 𝟒 generate the same sequence? Why might some people prefer this formula?
Yes. 𝟑(𝒏 − 𝟏) + 𝟒 = 𝟑𝒏 − 𝟑 + 𝟒 = 𝟑𝒏 + 𝟏. It is nice that the first term of the sequence is a term in the formula, so one
can almost read the formula in plain English: Since there is the “plus 3’” pattern, the nth term is just the first term plus
that many more threes.
c.
Graph the terms of the sequence as ordered pairs (𝒏, 𝒇(𝒏)) on the coordinate plane. What do you notice about the
graph?
The points all lie on the same line.
Module 3 Lesson1 Exit Ticket
1.
2.
Consider the sequence given by a “plus 8” pattern: 2, 10, 18, 26, …. Shae says that the formula for the sequence is
𝑓(𝑛) = 8𝑛 + 2. Marcus tells Shae that she is wrong because the formula for the sequence is 𝑓(𝑛) = 8𝑛 − 6.
a.
Which formula generates the sequence by starting at 𝑛 = 1? At 𝑛 = 0?
b.
Find the 100th term in the sequence.
Write a formula for the sequence of cube numbers: 1, 8, 27, 64, ….
Module 3 Lesson2 Exit Ticket
1.
2.
Consider the sequence following a “minus 8” pattern: 9, 1, −7, −15, ….
a.
Write an explicit formula for the sequence.
b.
Write a recursive formula for the sequence.
c.
Find the 38th term of the sequence.
Consider the sequence given by the formula 𝑎(𝑛 + 1) = 5𝑎(𝑛) and 𝑎(1) = 2 for 𝑛 ≥ 1.
a.
Explain what the formula means.
b.
List the first 5 terms of the sequence.
Module 3 Lesson3 Exit Ticket
1
Write the first 3 terms in the following sequences. Identify them as arithmetic or geometric.
c.
𝐴(𝑛 + 1) = 𝐴(𝑛) − 5 for 𝑛 ≥ 1 and 𝐴(1) = 9.
d.
𝐴(𝑛 + 1) = 𝐴(𝑛) for 𝑛 ≥ 1 and 𝐴(1) = 4.
e.
𝐴(𝑛 + 1) = 𝐴(𝑛) ÷ 10 for 𝑛 ≥ 1 and 𝐴(1) = 10.
2
c.
1
2
Identify each sequence as arithmetic or geometric. Explain your answer, and write an explicit formula for the
sequence.
1
a.
14, 11, 8, 5, …
b.
2, 10, 50, 250, …
3
5
7
− 2 , − 2 , − 2 , − 2, …