Doubling Pennies

Date: 11/26/96 at 06:41:26
From: Valerie
Subject: Accumulative math

I had to start with a penny, double it daily for 30 days, and know the 
total number for the month. I came up with 1,073,741,839.

I did 1+1=2+2=4+4=8+8...etc. then added them all up.  Was there an 
easier way?  Some formula?  While using a calculator, I figured out 
that I could take the answer times itself + 1 rather than typing the 
number twice.  It still took a long time, though.  Any suggestions?

Date: 11/26/96 at 08:16:40
From: Doctor Jerry
Subject: Re: Accumulative math

Valerie,

Your answer is very, very close to the answer I got, which is 
1,073,741,823.  Good work!

Here are two suggestions that will drastically reduce the amount of 
time needed for such calculations.

Many calculators have an "exponentiation" key. Sometimes it is marked 
with a a "y to the x" symbol. This symbol is written as the letter y 
and the letter x, with the letter x written just to the right and 
above the letter y. We often write that as y^x. The meaning is given 
by the following examples:

2^3 = 2*2*2 = 8
4^2 = 4*4 = 16.

y^x means to multiply y by itself x times.

So, if you want to calculate 2^(10), which is 2*2*2*...*2 (ten factors 
in all), your calculator should give 1,024.

Here's a way of thinking about the penny problem.

On day 1 you have 1 penney.

On day 2 you have 1+2 = 3 pennies. Notice that this is 1 less than 4 
pennies and 4 pennies is 2^2 = 2*2.

On day 3 you have 1+2+4 = 7 pennies. Notice that this is 1 less than 8 
pennies and 8 pennies is 2^3 = 2*2*2.

On day 4 you have 1+2+4+8=15 pennies. Notice that this is 1 less than 
16 pennies and 16 pennies is 2^4 = 2*2*2*2.

The pattern is very strong.  We can with confidence say that on day 30 
you will have 1 less than 2^(30) pennies.

My calculator gives 2^(30) = 1,073,741,824.

So the number of pennies is 1,073,741,824 - 1 = 1,073,741,823.

There's a puzzle question that's been around a long, long time. It's 
similar to your question. Here it is: If on the first square of a 
checkerboard you place 1 grain of wheat, on the second square you put 
2 grains of wheat, on the third square you put 4 grains of wheat, and 
continue this way, doubling the amount of wheat you put down on each 
square, will the amount of wheat required to fill all 64 squares of 
the checkerboard be the amount in a bushel, or the amount grown by one 
farmer in Kansas, or the amount grown by two farmers in Kansas, or 
something like the entire wheat production of Canada, or what?

When you take algebra, you will probably learn the formula for the sum 
of what is called a geometric progression.  

-Doctor Jerry,  The Math Forum
 Check out our web site!  http://mathforum.org/dr.math/