Exponents and Exponential Functions

Lesson 13

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).

Foundational Standards

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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.

Target Task

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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.