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# The Fractal Tree Revisited

Lesson 6 of 18

## Objective: SWBAT find the sum of an infinite geometric series.

*50 minutes*

#### Launch and Explore

*15 min*

During a previous lesson, we used a Fractal Tree to start investigating geometric sequences and series. Today we are going to** revisit this model** in order to make the connection from the partial sum of a geometric series to the sum of an infinite geometric series.

This worksheet will get students thinking about the** sum of an infinite geometric series **and whether or not they will increase with or without a bound. I give the worksheet to my students and have them work with their table groups on it for about 10 minutes.

Question #3 is really interesting and I will usually hear some **good conversations **about it. At this point I will pull the class back together and we will have a discussion about how long it will take for the fractal tree to reach a height of 5 units. Here are some thoughts that came up when I taught it:

- It will never reach a height of 5 units. The formula I came up for on the original day of the lesson for the height of the tree at stage
*n*is 4 - 1/2^(n-2), so whatever*n*is, we will be subtracting a small amount from 4 to get the height. The height will never even reach 4 units, let alone 5. - The height has to reach 5 units. We are adding a new unit to the height every time and the height is constantly increasing, so eventually it will have to reach 5 units.
- If we were to graph y = 4 – 1/2^(n-2), then 4 would be an asymptote. The graph would get really close to y = 4 but would never intersect the line.
- We are adding height to the tree with every stage, but the new height being added is getting really, really small. So eventually the new heights that are added will not affect the total height anymore.

It was really interesting to see those who originally thought that the height had to equal 5 units change their mind once they were presented with some really good arguments from those who disagreed. Their conversation was **a really good example of MP3** in practice!

#### Resources

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#### Extend

*25 min*

Once students have an intuitive sense of why the height of the fractal tree will never surpass 4, I ask them what will happen if we actually did add every single term of the infinite geometric series together. In the video below, I discuss how we can use the formula for the nth partial sum of a geometric series and modify it to find out the height of the fractal tree** if we added an infinite number of terms together**.

After we get this new formula, I present students with four sequences (shown here and here). I tell them to use this new formula for sequence B (where *r* = 2) and see what happens. They will get -1/2 as the infinite sum but clearly that cannot be correct since the terms are getting larger and larger and the sum will go to infinity.

Then I ask for some conjectures about when the sum will approach a certain number and when the sum will increase or decrease without bound. I introduce the vocabulary words **convergent and divergent** and explain their meaning to students. Next I give students a couple minutes to decide if the sum of the four geometric sequences will converge or diverge.

Usually my students will realize when 0 < *r* < 1 then the sum will definitely converge to a certain value. They are **unsure about negative r values**, so I encourage them to actually find partial sums until they figure it out. Once students figure this out, we will amend our formula for

*S*=

*a*/(1 –

*r*) to note that it only works when -1 <

*r*< 1.

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#### Summarize

*10 min*

Finally, I will assign students problems from our textbook that encompass all we have done with geometric sequences and series. I want to make sure that students work with partial sums and infinite sums of geometric series in this assignment. Also, I want to make sure that they are familiar with finding the *n*th term of a geometric sequence.

*expand content*

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- UNIT 1: Functioning with Functions
- UNIT 2: Polynomial and Rational Functions
- UNIT 3: Exponential and Logarithmic Functions
- UNIT 4: Trigonometric Functions
- UNIT 5: Trigonometric Relationships
- UNIT 6: Additional Trigonometry Topics
- UNIT 7: Midterm Review and Exam
- UNIT 8: Matrices and Systems
- UNIT 9: Sequences and Series
- UNIT 10: Conic Sections
- UNIT 11: Parametric Equations and Polar Coordinates
- UNIT 12: Math in 3D
- UNIT 13: Limits and Derivatives

- LESSON 1: The Skyscraper Problem
- LESSON 2: The Fractal Tree
- LESSON 3: Describing Sequences and Series
- LESSON 4: Arithmetic Sequences
- LESSON 5: Geometric Sequences
- LESSON 6: The Fractal Tree Revisited
- LESSON 7: Investments, Loans, and Mortgages - Day 1 of 2
- LESSON 8: Investments, Loans, and Mortgages - Day 2 of 2
- LESSON 9: Mathematical Induction
- LESSON 10: Formative Assessment Review: Sequences and Series
- LESSON 11: Formative Assessment: Sequences and Series
- LESSON 12: The Limit of a Sequence
- LESSON 13: Area Under a Curve - Day 1 of 2
- LESSON 14: Area Under a Curve - Day 2 of 2
- LESSON 15: Binomial Expansion
- LESSON 16: Unit Review: Sequences and Series
- LESSON 17: Unit Review Game: Pictionary
- LESSON 18: Unit Assessment: Sequences and Series