Geometric Sequences and Exponential Growth


I’m so glad Jacob brought up chain letters, and why haven’t I thought of that when I’m teaching about exponential growth?! I never participate in chain letters. They usually appeal to people’s desperate wish for more money, and I know they don’t work. But I got one that involved sending a book to a child, and had to write a nice letter to the parent who sent it to me. I don’t have the original. Here’s what I imagine it said (with changed names, of course):


Dear Sue, Your child can get lots of books!! Just send one! Buy one book and send it to the first child on the list below. Take that name off the list, and add your child’s name at the bottom. Send this on to six people. In a few weeks, your child will receive mail of their very own. Better than letters, it will be packages of books! 36 books, what a treat! You must send this on to 6 other parents soon, so you don’t break the chain. (If you don’t want to be part of this exciting adventure in books, please let me know as soon as you can. By the way, this is not illegal, as no money is being sent.)

1. Johnny Smith, 123 Unicorn Drive, Anytown, Ohio
2. Starla Jones, 456 Main St, Richmond, CA

Yours, Starla’s Mom


I wrote back (I’ve changed a few things to be clearer than I was at first):

Hi Starla’s Mom,

Thanks for thinking of us. Books are always a lovely idea. I’m including a book I picked up at a yard sale – I think Starla might like it. I wanted to get back to you quickly, as you requested, because I can’t do this. I never participate in chain letters because the chains can’t work for long. Let me tell you why. And if this chain works out for you against the odds, I’d love to hear about it.

You know I teach math; and I can’t help thinking of this as a math problem… If you suppose that the letter was dug up from an old stack by the family of the first child on your list, then she or he would be the first layer (in the current cycle). The second child on your list would be in the second layer (and there would be 6 children in that layer). Starla would be in the 3rd layer, among 36 (6x6) children. My son would be in the 4th layer, among 216 (6x6x6) children. He would get books from children in the 6th layer, which totals 7,776 (6 to the 5th power) children. The children in that layer would get books from children in the 8th layer, which totals 279,936 children (6 to the 7th). They get books from kids in the 10th layer, which totals 10,077,696 children. They won’t have enough kids to give them books, because the 12th layer has 362,797,056 children in it, more than the whole population of the U.S. I just googled: there are just over 70 million kids in the U.S. If everyone did it, they’d almost all get this letter by the 11th layer (60 million in that layer, plus all the previous). So the kids up to the 9th layer would get their books, but the last 2 layers wouldn’t. 2 million kids would get books, at the expense of 70 million who’d be disappointed. Doesn’t seem fair to me… (And, of course, the chances that we’re already in the disappointing stages of this game are the same … 34 out of 35, or about 97%. I sure don’t want to disappoint my boy.)

If you’d like to do a book exchange that doesn’t use exponential growth, maybe we could figure something out together. It might be fun to exchange books among multiracial families, or adoptive families, or ...

Warmly,
Sue



The previous lesson was about arithmetic sequences. You added the same number each time. (In this lesson, on geometric sequences, you multiply by the same thing each time.) If you start with an arithmetic sequence, and make a different sequence by writing terms that are each one bigger than the terms of the original, do you still get an arithmetic sequence?

Example: 1 4 7 10 … is arithmetic, with a common difference of 3. If you make each term one bigger, you’d have 2 5 8 11 …, which still has a common difference of 3.

Here’s a related problem from the 1st edition that’s missing in the 3rd:
If one is added to each term of a geometric sequence, is the resulting sequence also geometric? Try it out with one and see.

When we get to the next chapter on functions and their graphs, we’ll see that arithmetic sequences are pretty similar to linear functions; they both model situations in which something grows the same amount in each unit of time, for example. And geometric sequences are similarly related to exponential functions; they both model situations in which something grows by the same percentage in each unit of time. (Exponential functions are covered at the end of chapter 4 in the 3rd edition.)

Here’s one more problem I’ve enjoyed, which uses the ideas in these sections:
Sally’s mom offers her two different pay scales (for digging out weeds to make a new garden bed) for the next 30 days.
Plan A: $10 on day 1, $20 on day 2, $30 on day 3, $40 on day 4, and so on, or
Plan B: 1 cent on day 1, 2 cents on day 2, 4 cents on day 3, 8 cents on day 4, and so on.

Which plan would you choose? Is each of these plans an arithmetic sequence, a geometric sequence, or neither? What would be the total Sally would earn with each plan?

Have fun!

- Notes by Sue VanHattum