# Exponents and Exponential Functions

## Objective

Write recursive formulas for sequences, including the Fibonacci sequence.

## Common Core Standards

### Core Standards

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• 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.

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• 8.F.B.4

## Criteria for Success

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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

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### 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 ${a_{14}}$?

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

### 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.

## Problem Set

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

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### 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..