Infinite Geometric Series
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1 Infinite Geometric Series Grade: 11 Subject: Pre- Calculus 11 Unit: Sequences and Series Driving Question: If a sequence has an infinite number of terms, is it possible to determine its sum? Curriculum Outcomes: GCO s Students will be expected to develop algebraic and graphical reasoning through the study of relations. SCO s RF10: Analyze geometric sequence and series to solve problems. Expected Time: 2 classes Resources: (Tools & Tech) Lesson Procedure I Do: Lead demonstration activity. In this activity, students will divide and subdivide a piece of paper. A class discussion will result as students predict and debate what will happen to the paper and how to determine the sum of the parts. You Do: Have students complete both parts of individual activity. This activity will look at two instances where the sum of an infinite series can be found and look at what specific characteristics make it possible to find the sum. I Do:
2 Screencast Video Go over worksheets and ensure that students understand the principle that!! = 0. Develop the formula for an infinite geometric series and explain the parameters that are required for it to be applicable. Screencast video available for review of formula and how it works. EeUsUE We Share: Have students work in groups to research and explore situations where infinite geometric series exist in the real world. Each group should be prepared to present at least one example to the group as a whole. Student Worksheets: Infinite Geometric Series
3 INFINITE GEOMETRIC SERIES Demonstration Activity: Start with one full sheet of paper. Select three students to participate, one person is the owner of the paper, the other two are going to be recipients. The owner is to share the paper equally by splitting it in thirds and giving each of the recipients one third of the paper. At this point, everyone should now have an equal amount of paper. Next, the two recipients ask for more paper, an equal share of what the owner has left. The owner is to split the remaining piece in thirds again and share. Repeat this process as long as possible. Will the owner ever be completely out of paper? Why or why not? What fraction of the paper do each of the other two people have? What happens when you add an infinite number of terms together? You might think that no matter how small they are, if you sum infinitely, the answer will go to infinity. In some cases this is true, however there are some cases where the final answer is a finite real number. Explain why you think this might be true.
4 INFINITE GEOMETRIC SERIES Part One: Given the line segment AB: A B Determine the length of the entire segment AB? Find the midpoint of AB, label it M. Write the length of AM in the space above it. Find the midpoint of MB, label it M. Write the length of MN in the space above it. Find the midpoint of NB, label it P. Write the length of NP in the space above it. Find the midpoint of PB, label it Q. Write the length of PQ in the space above it. Find the midpoint of QB, label it R. Write the length of QR in the space above it. Find the midpoint of RB, label it S. Write the length of RS in the space above it. Write the lengths of the values found above as a sequence. Predict the next 3 terms of this sequence. Is this sequence arithmetic, geometric, or neither? Explain how you know. Explain whether or not this sequence could continue infinitely?
5 Part Two: Given the following square (assume area is 1 unit 2 ), follow the directions below using a ruler and coloured pencils: For each of the following steps, choose a different colour. Draw and shade a rectangle with an area of ½ unit 2. Draw and shade another rectangle, filling half of the remaining open space. Repeat the process until you are no longer able to create a rectangle.
6 1. Identify the area of each rectangle and write the area of the entire square as a sum. 2. What is the sum of the series? 3. Is this an example of an infinite series? Why or why not? 4. What do the sequences in both Part One and Part Two have in common? 5. Explain what happens to the value of the n th term as n becomes very large.
7 Solution:!! +!! +!! +!!" +!!" +!!" +!!"# +!!"# + = 1
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