Infinite Geometric Series

I did a bad job (in my opinion) of teaching infinite geometric series in precalculus in my previous class. I told them I did a bad job. I was rushing. They were confused. (One of them said: “you did a fine job, Mr. Shah” which made me feel better, but I still felt like they were super confused.)

At the start of the lesson, I gave each group one colored piece of paper. (I got this idea last year from my friend Bowen Kerins on Facebook! He is not only a math genius but he’s also a 5 time world pinball player champion. Seriously.) I don’t know why but it was nice to give each group a different color piece of paper. Then I had them designate one person to be the “paper master” and two people to be the friends of the paper master. Any group with a fourth person simply had to have the fourth person be the observer.

I did not document this, so I have made photographs to illustrate ex post facto.


I started, “Paper master, you have a whole sheet of paper! One whole sheet of paper! And you have two friends. You feel like being kind, sharing is caring, so why don’t you give them each a third of your paper.”

The paper master divided the paper in thirds, tore it, and shared their paper.


Then I said: “Your friends loveeeed their paper gift. They want just a little bit more. Why don’t you give them each some more… Maybe divide what you have left into thirds so you can keep some too.”

And the paper master took what they had, divided it into thirds, and shared it.


To the friends, I said: “Hey, friends, how many of you LOOOOOVE all these presents you’re getting? WHO WANTS MORE?” and the friends replied “MEEEEEEEEEEEEEEE!”

“Paper master, your friends are getting greedy. And they demand more paper. They said you must give them more or they won’t be your friends. And you are peer pressured into giving them more. So divide what little you have left and hand it to them.”

They do.


“Now do it again. Because your greedy friends are greedy and evil, but they’re still your friends.”






Here we stop. The friends have a lot of slips of paper of varying sizes. The paper master has a tiny speck.

I ask the class: “If we continue this, how much paper is the paper master going to eventually end up with?”

(Discussion ensues about whether the answer is 0 or super duper super close to 0.)

I ask the class: “If we continue this, how much paper are each of the friends going to have?”

(A more lively short discussion ensues… Eventually they agree… each friend will have about 1/2 the paper, since there was a whole piece of paper to start, each friend gets the same amount, and the paper master has essentially no paper left.)

I then go to the board.

I write \frac{1}{2}=

and then I say: “How much paper did you get in your initial gift, friends?”

I write \frac{1}{2}=\frac{1}{3}+

and then we continue, until I have:


Ooohs and aahs.

Next year I am going to task each student to do this with two friends or people from their family, and have them write down their friends/family member’s reactions…

I love this.



  1. Did this a couple years ago with a mixed group (mid-elementary to teen) of homeschoolers, only it wasn’t “paper” it was cake, and Mom was sharing it out. But she was “on a diet”, so she kept deciding to cut up her piece and give each of the kids just a little bit more. Their math skills weren’t up to extending this to a general geometric series, but for the 1/n type of series, it was easy to see that no matter how many people we started with, each child always got 1/(n-1) of the cake as Mom’s piece dwindled to zero.

    And we also enjoyed the discussion of whether you could ever cut cake crumbs all the way to zero :)

    My daughter (now 9th grade) still remembers the “infinite cakes” and occasionally uses them in a math problem—for instance, to find the height of the triangle in this problem:

    Click to access MAA_AMC_Inspiration_Letter-1_2012.pdf

  2. Haha. When I got to the end, I literally said, “Ooooh!” And then I read that your students said, “Oooh and Aaaah” as well and I laughed. I love it. A good lesson is like setting off mathematical fireworks in the brain. And what’s a good fireworks show without some Ooohs and Aaahs. Way to put on a good show! :)

  3. Sam
    I cannot see any of the images. They are blank spaces and when I click on them I am told that I denied access. Could this be a browser issue? I’m using Safari.

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