Math Doubling

Let’s start with an easier problem.  Let’s assume you have a penny and each day the amount you have doubles.  Listing the amount you have each day would look like this: You already had the 1 penny.
(1)   2   4     8     16     32     64…….  You could also write this as
2°   2¹  2²   2³   ……..  You would see that for the first day of doubling you have two to the first power… and exponent goes up by one each day.  After 100 days you’d have 2 to the 100th power. That is a lot. It comes to     1267650600228229401496703205376 pennies. You now understand why I suggested you not waste your time doing the actual calculations.

Let’s make some sense of this. You can move the decimal to the left two places and have dollars. That is still too much to make sense of. We have a hard time perceiving millions, billions and trillions. Moving the decimal 12 more places to the left would give an answer in the trillions of dollars.  You would have over 1267650600228229 trillion dollars. Go back another 12 decimal points and you would have 1,267 trillion trillions.

But remember that was an easier question. All you needed to see was that you’d have a total of 2 to the 100th power of pennies.

Now lets look at the actual problem.  You have been give one penny. Each day the amount you are given doubles. On the one hundredth day you’d be given 2 to the 100th power of pennies in addition to everything you were given on the first 99 days. I had my students do a similar problem for extra credit but only doubling for 30 days. I thought this step would take hours. One smart student solved it in minutes. Can you? The daily amounts start as
given to you  (1)    2    4       8    16     32
Your total       1      3     7    15    31     63

When it is written this way the answer should be easy to see. Look at the numbers until you see the pattern.

You would have  about 2534  trillion trillions of dollars. With smaller numbers, I would subtract 1 but I haven’t even rounded off so I’ve already subtracted many trillions of dollars. I won’t worry about subtracting 1.

Doing math can be boring, but many of us believe finding patterns like this one is fascinating.  If you still don’t see the pattern either ask a math major, a math teacher or email me.