Infinite Geometric Sequences and Series (Lesson 7.3 Day 2)
Unit 7 - Day 7
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
Quick Lesson Plan
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.
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.