# Infinite Geometric Sequences and Series (Lesson 7.4)

###### Unit 0: Prerequisites

**Day 1:**The Cartesian Plane**Day 2:**Equations of Circles**Day 3:**Solving Equations in Multiple Representations**Day 4:**Reasoning with Formulas**Day 5:**Quiz 0.1 to 0.4**Day 6:**Linear Relationships**Day 7:**Reasoning with Slope**Day 8:**Set Notation**Day 9:**Quiz 0.5 to 0.7**Day 10:**Unit 0 Review**Day 11:**Unit 0 Test

###### Unit 1: Functions

**Day 1:**Functions and Function Notation**Day 2:**Domain and Range**Day 3:**Rates of Change and Graph Behavior**Day 4:**Library of Parent Functions**Day 5:**Transformations of Functions**Day 6:**Transformations of Functions**Day 7:**Even and Odd Functions**Day 8:**Quiz 1.1 to 1.6**Day 9:**Building Functions**Day 10:**Compositions of Functions**Day 11:**Inverse Functions**Day 12:**Graphs of Inverse Functions**Day 13:**Piecewise Functions**Day 14:**Quiz 1.7 to 1.11**Day 15:**Unit 1 Review**Day 16:**Unit 1 Test

###### Unit 2: Polynomial and Rational Functions

**Day 1:**Connecting Quadratics**Day 2:**Completing the Square**Day 3:**Polynomials in the Short Run**Day 4:**Polynomials in the Long Run**Day 5:**Review 2.1-2.4**Day 6:**Quiz 2.1 to 2.4**Day 7:**Factor and Remainder Theorem**Day 8:**Factor and Remainder Theorem**Day 9:**Complex Zeros**Day 10:**Connecting Zeros Across Multiple Representations**Day 11:**Intro to Rational Functions**Day 12:**Graphing Rational Functions**Day 13:**Quiz 2.5 to 2.9**Day 14:**Unit 2 Review**Day 15:**Unit 2 Test

###### Unit 3: Exponential and Logarithmic Functions

**Day 1:**Exponential Functions**Day 2:**Graphs of Exponential Functions**Day 3:**Compound Interest and an Introduction to "e"**Day 4:**Review 3.1-3.3**Day 5:**Quiz 3.1 to 3.3**Day 6:**Logarithmic Functions**Day 7:**Graphs of Logarithmic Functions**Day 8:**Logarithm Properties**Day 9:**Solving Exponential and Logarithmic Equations**Day 10:**Quiz 3.4 to 3.7**Day 11:**Exponential and Logarithmic Modeling**Day 12:**Unit 3 Review**Day 13:**Unit 3 Test

###### Unit 4: Trigonometric Functions

**Day 1:**Right Triangle Trig**Day 2:**Inverse Trig Ratios**Day 3:**Radians and Degrees**Day 4:**Unit Circle**Day 5:**Unit Circle**Day 6:**Other Trig Functions**Day 7:**Review 4.1-4.6**Day 8:**Quiz 4.1 to 4.6**Day 9:**Graphing Sine and Cosine**Day 10:**Transformations of Sine and Cosine Graphs**Day 11:**Graphing Secant and Cosecant**Day 12:**Graphing Tangent and Cotangent**Day 13:**Quiz 4.7 to 4.10**Day 14:**Inverse Trig Functions**Day 15:**Trigonometric Modeling**Day 16:**Trigonometric Identities**Day 17:**Unit 4 Review**Day 18:**Unit 4 Review**Day 19:**Unit 4 Test

###### Unit 5: Applications of Trigonometry

**Day 1:**Law of Sines**Day 2:**The Ambiguous Case (SSA)**Day 3:**Law of Cosines**Day 4:**Area and Applications of Laws**Day 5:**Vectors**Day 6:**Review 5.1-5.5**Day 7:**Quiz 5.1 to 5.5**Day 8:**Polar Coordinates**Day 9:**Equations in Polar and Cartesian Form**Day 10:**Polar Graphs Part 1**Day 11:**Polar Graphs Part 2**Day 12:**Review 5.6-5.9**Day 13:**Quiz 5.6 to 5.9**Day 14:**Parametric Equations**Day 15:**Parametric Equations (With Trig)**Day 16:**Unit 5 Review**Day 17:**Unit 5 Test

###### Unit 6: Systems of Equations

**Day 1:**What is a Solution?**Day 2:**Solving Systems with Substitution**Day 3:**Solving Systems with Elimination**Day 4:**Review 6.1-6.3**Day 5:**Quiz 6.1 to 6.3**Day 6:**Solving Systems in 3 Variables**Day 7:**Solving Systems in 3 Variables**Day 8:**Partial Fractions**Day 9:**Unit 6 Review**Day 10:**Unit 6 Test

###### Unit 7: Sequences and Series

**Day 1:**Introducing Sequences**Day 2:**Using Sequences and Series to Describe Patterns**Day 3:**Arithmetic Sequences and Series**Day 4:**Review 7.1-7.2**Day 5:**Quiz 7.1 to 7.2**Day 6:**Geometric Sequences and Finite Series**Day 7:**Infinite Geometric Sequences and Series**Day 8:**Proof by Induction**Day 9:**Proof by Induction**Day 10:**Quiz 7.3 to 7.5**Day 11:**Unit 7 Review**Day 12:**Unit 7 Test

###### Unit 8: Limits

**Day 1:**What is a Limit?**Day 2:**Evaluating Limits Graphically**Day 3:**Evaluating Limits with Direct Substitution**Day 4:**Evaluating Limits Analytically**Day 5:**Evaluating Limits Analytically**Day 6:**Review 8.1-8.4**Day 7:**Quiz 8.1 to 8.4**Day 8:**Continuity**Day 9:**Continuity**Day 10:**Intermediate Value Theorem**Day 11:**Intermediate Value Theorem**Day 12:**Review 8.5-8.6**Day 13:**Quiz 8.5 to 8.6**Day 14:**Limits at Infinity**Day 15:**Unit 8 Review**Day 16:**Unit 8 Test

###### Unit 9: Derivatives

**Day 1:**Introduction to Derivatives**Day 2:**Average versus Instantaneous Rates of Change**Day 3:**Calculating Instantaneous Rate of Change**Day 4:**Calculating Instantaneous Rate of Change**Day 5:**The Derivative Function**Day 6:**The Derivative Function**Day 7:**Review 9.1-9.3**Day 8:**Quiz 9.1 to 9.3**Day 9:**Derivative Shortcuts**Day 10:**Differentiability**Day 11:**Connecting f and f’**Day 12:**Connecting f and f’**Day 13:**Review 9.4-9.6**Day 14:**Quiz 9.4 to 9.6**Day 15:**Derivatives of Sine and Cosine**Day 16:**Product Rule**Day 17:**Quotient Rule**Day 18:**Review 9.7-9.9**Day 19:**Quiz 9.7 to 9.9**Day 20:**Unit 9 Review**Day 21:**Unit 9 Test

###### Unit 10: (Optional) Conic Sections

**Day 1:**Intro to Conic Sections**Day 2:**Defining Parabolas**Day 3:**Working with Parabolas**Day 4:**Quiz 10.1 to 10.3**Day 5:**Defining Ellipses**Day 6:**Working with Elllipses**Day 7:**Defining Hyperbolas**Day 8:**Working with Hyperbolas**Day 9:**Quiz 10.4 to 10.7**Day 10:**Unit 10 Review**Day 11:**Unit 10 Test

#### Learning Targets

Explore the behavior of a geometric sequence as n approaches infinity.

Understand when and how adding infinitely many terms can lead to a finite sum.

Tasks/Activity | Time |
---|---|

Activity | 20 minutes |

Debrief Activity | 10 minutes |

Important Ideas | 5 minutes |

Check Your Understanding | 15 minutes |

#### Activity: How Does the Chicken Cross the Road?

#### Lesson Handouts

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#### Answer Key

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#### Homework

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#### Experience First

We often ask why the chicken crossed the road, but today we explore an even more interesting question: *how* did the chicken cross the road? Get ready for heated arguments and mind-boggled students as your class explores Bernie’s path across a 12 foot road.

Question 1 and 2 are pretty straightforward for students and provide an easy entry point to the lesson. Check to make sure students are writing the nth term and not the nth sum in the table (how far Bernie walks on each leg, not how far he’s walked in total).

In question 3, students use a graphical approach to explore what happens to Bernie’s total distance. Almost every group concluded that since there is a horizontal asymptote at y=12, Bernie will never cross the road! Be ready to play devil’s advocate here. I ask my students: “How do we make it anywhere then? How do we even make it to the door? Don’t I have to get half-way to the door first? And then don’t I have to make it half of the remaining distance? You’re telling me I’ll never make it to the door?!” The students are quick to say that humans don’t walk like Bernie does, and we step over the half-way mark. I then bring up a marble that is rolling on the floor. Surely the marble doesn’t skip half-way marks?!

Still students weren’t all that convinced. We further the argument in question 5. Students first say that it will take him infinitely long, but upon further inspection they realize it should take him 128 seconds, since the second 6 feet should take him just as long as the first six feet. At this point many still think he won’t cross the road but this is an obvious paradox. Ask students whether he will cross the road in 128 seconds or whether he will never cross the road, because it can’t be both. Walk away when you pose these advancing questions and allow students to wrestle further.

In question 7, students are able to reason that adding any amount of pause would make him not arrive, since an infinite amount of equal length pauses would take an infinite amount of time. This is a key distinction between infinite arithmetic sequences (and infinite geometric series with a common ratio greater than or equal to 1) and infinite geometric sequences with a common ratio less than 1.

#### Formalize Later

Students are excited for the debrief because they are honestly searching for some resolution about this weird chicken! The key ideas to bring up are that even though it would require infinitely many legs of the journey to cross the road, the total distance (sum) is finite. Pause to consider how strange this concept is even for us who have been teaching this content for a while! One student said it nicely when she explained that “even though there are infinitely many halves, they only add up to a fixed amount because the amount being added is getting smaller and smaller”. This is a great introduction to the idea of limits, though we save the formal notation for next chapter, opting instead to use the more informal arrows. As n (the number of legs) goes to infinity, S(n) (the total distance traveled) goes to 12. A similar argument can be made for the time it takes to complete the journey.

In the Important Ideas, we use a flow chart to show the two cases of infinite geometric sums. We specifically use the language of “does not exist” since this will prepare students for limits at infinity in the next unit.

Although the lesson doesn’t go into great detail about why the infinite sum formula is a(1)/(1-r), your class may be ready for a short conversation about how this formula relates to the finite sum formula they learned yesterday. Note that if 0<r<1, then as n goes to infinity, r^n gets closer and closer to 0 which means 1-r^n gets closer and closer to 1. This is why the formula only has a(1) in the numerator.