Doubling Sequence

Date: 8/24/96 at 11:46:54
From: casyfarm
Subject: Doubling Sequence

Dr. Math,

Very far away there was a country with unusual weather conditions.  
One winter the following occurred:  Every day in January it snowed.  
On Jan. 1st it snowed 1 cm.  On Jan. 2, it snowed 2 cm. On Jan. 3, 
it snowed 4 cm. On Jan. 4, it snowed 8 cm, and so on until Jan. 31.  
How much snow fell on Jan. 31st?  Was it more than the length of a 
football field?

I figured out the sequence and can figure out the answer by doubling 
the snowfall every day. There must be a shorter way of doing this.  I 
figure the answer is 2 to the 30th power, but can't think of a formula 
to get the answer.  Can you help?  I know to convert cm to yards: 
you divide by 91.44.  

Thank you,
Maureen Casey

Date: 8/25/96 at 16:2:26
From: Doctor Robert
Subject: Re: Doubling Sequence

The formula for finding the sum of a geometric sequence is fairly 
difficult to derive, but I can tell you what it is.  The sum of the 
first n terms is 

  S = (first term)*(1 - r^n)/(1-r)

In your case the first term is 1, the ratio, r, is 2 (the amount of 
snow doubles every day) and n (the number of terms) is 31.

If you plug these numbers into a calculator you get a tremendous 
amount of total snow - more than 13,000 miles of it.  

Try talking your parents into giving you an allowance of 1 penny, 2 
pennies, 4 pennies, etc.  for a month.  You'll be a multi-millionaire!

It is fairly easy to figure out how much snow fell on _just_ Jan 31. 
Let's look at the sequence, and look for a pattern:

Day:      Amount it snowed:
Jan 1     1 cm
Jan 2     1 * 2  = 2 cm
Jan 3     1 * 2 * 2  = 4 cm
Jan 4     1 * 2 * 2 * 2  = 8 cm
...       ...

Do you see the pattern? For each day after the first, there is another 
doubling of the previous day's snowfall. By the 31st, the snowfall has 
doubled 30 times. 

To double something 30 times, you can either multiply 2*2*2... 
thirty times, or write this as 2^30 (two raised to the thirtieth 
power). Most calculators can solve 2^30 for you (it's a really big 
number!).

Therefore, your answer is that 2^30 cm of snow fell on the 31st. 
You can divide 2^30 to figure out how many yards that is.

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