# Exponents and Exponential Functions

## Objective

Define arithmetic and geometric sequences, and identify common ratios and common differences in sequences.

## 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.LE.A.2 — Construct linear and exponential functions, including arithmetic and geometric sequences, given a graph, a description of a relationship, or two input-output pairs (include reading these from a table).

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• 8.F.B.4

## Criteria for Success

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1. Define arithmetic sequences as those that increase or decrease linearly and have a common difference (constant average rate of change) between terms.
2. Define geometric sequences as those that increase or decrease exponentially and have a common ratio (increasing/decreasing average rate of change) between terms.
3. Identify the common difference or common ratio for arithmetic and geometric sequences, respectively.
4. Write recursive formulas for arithmetic and geometric sequences.

## Anchor Problems

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### Problem 1

Four sequences are shown below. Which one does not belong and why?

 ${4, 8, 12, 16, 20 ...}$ ${50, 46, 42, 38, 34, ...}$ ${1, 4, 16, 64, 216, ...}$ ${1600, 400, 100, 25, 12.5, ...}$

#### Guiding Questions

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### Problem 2

Complete each sequence by filling in the missing terms. Then identify each sequence as arithmetic or geometric and name the common difference or common ratio.

a.   ${-2}$, $4$, ___, ___, ___, ${28}$, ...

b.   ${{1\over2}}$, ___, ___, $4$, $8$, ___, …

c.   $5$, ___, ___, ___, ${2{1\over3}}$, ${1{2\over3}}$, ...

d.   ___, $3$, ${-9}$, ___, $-81$, ___, …

#### Guiding Questions

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### Problem 3

Write a recursive formula for each sequence.

a.   ${11, \space14,\space 17,\space 20,\space 23, …}$

b.   ${{1\over9},\space {1\over3},\space1,\space 3,\space 9,\space …}$

c.   ${32,\space 16,\space 8,\space 4,\space 2,\space …}$

#### Guiding Questions

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## 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

Write the first three terms in the following sequences. Identify them as arithmetic or geometric.

1. ${A(n+1)=A(n)-5}$ for ${{{n\geq1}}}$ and ${A(1)=9}$
2. ${A(n+1)={1\over2}A(n)}$ for ${{{n\geq1}}}$ and ${A(1)=4}$
3. ${A(n+1)=A(n)\div10}$ for ${{{n\geq1}}}$ and ${A(1)=10}$

#### References

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

Algebra I > Module 3 > Topic A > Lesson 3 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

Identify each sequence as arithmetic or geometric. Explain your answer and write a recursive formula for the sequence.

a.  ${14,11,8,5,…}$

b.  ${2,10,50,250,…}$

c.  ${-{1\over2}, -{3\over2}, -{5\over2},-{7\over2},...}$

#### References

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

Algebra I > Module 3 > Topic A > Lesson 3 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..

Modified by The Match Foundation, Inc.