Doubling SequenceDate: 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/ |
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