Exponents and Exponential Functions

Lesson 12

Objective

Write recursive formulas for sequences, including the Fibonacci sequence.

Common Core Standards

Core Standards

?

  • F.BF.A.2 — Write arithmetic and geometric sequences both recursively and with an explicit formula, use them to model situations, and translate between the two forms. Modeling is best interpreted not as a collection of isolated topics but in relation to other standards. Making mathematical models is a Standard for Mathematical Practice, and specific modeling standards appear throughout the high school standards indicated by a star symbol (★). The star symbol sometimes appears on the heading for a group of standards; in that case, it should be understood to apply to all standards in that group.

  • F.IF.A.2 — Use function notation, evaluate functions for inputs in their domains, and interpret statements that use function notation in terms of a context.

  • F.IF.A.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.

Foundational Standards

?

  • 8.F.B.4

Criteria for Success

?

  1. Define the Fibonacci sequence and represent it recursively. 
  2. Represent a sequence with a recursive formula, identifying the relationship between terms and defining the value of the first term. 
  3. Write a sequence given an explicit formula.

Anchor Problems

?

Problem 1

Consider the sequence below.

$${1, \space1,\space 2,\space 3,\space 5,\space 8,\space 13,\space 21, \space34, …}$$

a.   Describe the pattern that you notice. How is each next term determined?

b.   This pattern is famously called the Fibonacci sequence. Write a recursive formula to represent the Fibonacci sequence. Write a formula in sequence notation using $${a_n}$$ and in formula notation using $${f(n)}$$.

c.   If $${a_{15}=610}$$ and $${a_{16}=987}$$, what are the values of $${a_{17}}$$ and $${a_{14}}$$?

Guiding Questions

Create a free account or sign in to access the Guiding Questions for this Anchor Problem.

Problem 2

For each sequence below, write a recursive formula, in both sequence notation and function notation, to represent the sequence. 

a.   $${18,\space 13,\space 8,\space 3,\space -2,\space …}$$

b.   $${4,\space 12,\space 36,\space 108,\space 324,\space …}$$

Guiding Questions

Create a free account or sign in to access the Guiding Questions for this Anchor Problem.

Problem 3

A sequence is given by an explicit formula, as shown below, where $$n$$ represents the term number. 

$$a_n=4n-1$$ for $$n\geq1$$

a.   Write the first five terms in the sequence.

b.   Write a recursive formula to represent the sequence. 

Guiding Questions

Create a free account or sign in to access the Guiding Questions for this Anchor Problem.

Problem Set

?

The following resources include problems and activities aligned to the objective of the lesson that can be used to create your own problem set.

Target Task

?

Problem 1

Consider the sequence following a minus $$8$$ pattern: $${9,\space1,\space-7,\space-15\space,...}$$

Write a recursive formula for the sequence.

References

EngageNY Mathematics Algebra I > Module 3 > Topic A > Lesson 2Exit Ticket, Question #1b

Algebra I > Module 3 > Topic A > Lesson 2 of the New York State Common Core Mathematics Curriculum from EngageNY and Great Minds. © 2015 Great Minds. Licensed by EngageNY of the New York State Education Department under the CC BY-NC-SA 3.0 US license. Accessed Dec. 2, 2016, 5:15 p.m..

Problem 2

Consider the sequence given by the formula $${a(n+1)=5a(n)}$$ and $${a(1)=2}$$ for $${n\geq1}$$

  1. Explain what the formula means.
  2. List the first five terms of the sequence. 

References

EngageNY Mathematics Algebra I > Module 3 > Topic A > Lesson 2Exit Ticket, Question #2

Algebra I > Module 3 > Topic A > Lesson 2 of the New York State Common Core Mathematics Curriculum from EngageNY and Great Minds. © 2015 Great Minds. Licensed by EngageNY of the New York State Education Department under the CC BY-NC-SA 3.0 US license. Accessed Dec. 2, 2016, 5:15 p.m..