Part IV: Shifting into High Gear with Advanced Concepts in VS .NET

Generation QR-Code in VS .NET Part IV: Shifting into High Gear with Advanced Concepts

n 1 2 3 4 5 6

3 1 7 + = 2 4 4 7 1 15 + = 4 8 8 15 1 31 + = 8 16 16 31 1 63 + = 16 32 32 4,095 2,048 2 (2 n 1 ) 1 = 2 n 1 =

12 n

Having the tools to add all the terms in a mathematical sequence is peachy keen from a homework standpoint, but what s the point outside the classroom Why would anyone need to be able to add sequences in the real world You may be surprised at the possible applications in many walks of life Hopefully, the three examples I include in this section will give you a hint of how helpful figuring sums can be

You ve been hired for cleanup duty in a huge theater after a performance by a popular band One of your tasks is to steam-clean all the seats on the main floor Because it takes two minutes per seat, you need to allocate enough time to do the entire job The head of maintenance tells you that the first row has 36 seats, and each subsequent row has one more seat per row The theater has a total of 25 rows How many seats are in the theater How much time will it take

This problem calls for the sum of an arithmetic sequence (see the section Summing arithmetically ) The first term is 36, the common difference is 1, and you have 25 terms Using the formula for the sum of the terms of an arithmetic sequence, you calculate the following: s n = n 8 2a 1 + ^ n - 1h d B 2 25 2 36 + 25 - 1 1 = h$ B 8 ^ h ^ 2 = 25 6 72 + 24 @ 2 25 96 = 1, 200 = 6 @ 2 You have 1,200 seats to steam-clean At 2 minutes per seat, you need to set aside 2,400 minutes, or 40 hours for the job Better hire some help!

You approach your dad about increasing your allowance because $10 per week just doesn t cut it anymore He replies, Absolutely not Not until you improve your math grade You then negotiate a deal with the following proposition: You ll take 1 cent on the first day of the month, 2 cents on the second day of the month, 4 cents on the third, 8 cents on the fourth, and so on, doubling the amount each day until the end of the month At that point, your dad can check on your math grade and see if he wants to change the system back and raise your allowance He knows how crafty you are, so he asks you to explain what you re up to before he agrees How much will you get with your system for the month of January That month has 31 days of penny-pinching allowance Using the formula for the sum of a geometric sequence whose first term is 1, common ratio is 2, and number of terms is 31 (see the section Summing geometrically ), you calculate the following: Sn= a 1_1 - r n i 1-r 1 _1 - 2 31i = 1-2 1 - 2,147, 483, 648 = -1 = 2,147, 483, 647

Of course, your answer is in pennies, so you move the decimal point over Your allowance comes to a total of $21,474,83647 What does your dad think of your math ability now Unfortunately, he wants you to care about your grades as much as your money No deal