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Unformatted text preview: 6.042/18.062J Mathematics for Computer Science March 10, 2005 Srini Devadas and Eric Lehman Lecture Notes Sums and Approximations When you analyze the running time of an algorithm, the probability some procedure succeeds, or the behavior of a load-balancing or communications scheme, youll rarely get a simple answer. The world is not so kind. More likely, youll end up with a complicated sum: n 1 k + k k =1 Or a nasty product: 1 2 3 n 1 + 1 + 1 + 1 + 2 2 2 2 n n n n Or you might get an answer that is just tad too complicated to grasp intuitively: 72 /n n 1 + 100 And these examples are relatively benign! So the ability to cope with such messy mathematical expressions is an important skill in computer science and in many other areas of science and engineering. This might not seem glorious, but people who can cut monstrous expressions down to size in moments can seem pretty amazing to the uninitiated. This week, well equip you with the most useful tricks of the trade. 1 The Value of an Annuity Would you prefer a million dollars today or $20,000 a year for the next fifty years? This is a question about the value of an annuity , a financial instrument that pays out a fixed amount of money at the beginning of every year for some specified number of years. In particular, an n-year, $ m-payment annuity pays m dollars at the start of each year for n years. In some cases, n is finite, but not always. Examples include lottery payouts, student loans, and home mortgages. There are even Wall Street people who specialize in trading annuities. For many reasons, $20,000 a year for 50 years is worth much less than a million dollars right now. For example, consider the last $20,000 installment. If you had that $20,000 right now, then you could start earning interest, invest the money in the stock market, 2 Sums and Approximations or just buy something fun. However, if you dont get the $20,000 for another 50 years, then someone else is earning all the interest or investment profit. Furthermore, prices are likely to gradually rise over the next 50 years, so you when you finally get the money, you wont be able to buy as much. Finally, people only live so long; if you were 60 years old, a payout 50 years in the future would be worth next to nothing! But what if your choice were between $40,000 a year for 50 years and a million dollars today? Now which is better? What is an annuity is actually worth? 1.1 The Future Value of Money In order to address such questions, we have to make an assumption about the future value of money. Lets put most of the complications aside and think about this from a simple-minded perspective. The average rate of ination in the United States from 1980 to 2004 was about p = 3 . 5% per year. This means that the price of a selection of basic goods increases by about 3.5% each year. If this trend continues, then goods costing $100 today will cost: $100(1 + p ) = $103 . 50 in 1 year $100(1 + p ) 2 = $107 . 12 in 2 year...
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This note was uploaded on 02/08/2011 for the course EECS 6.042 taught by Professor Dr.ericlehman during the Spring '11 term at MIT.
- Spring '11
- Computer Science