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

Creator QR Code in Visual Studio .NET Part IV: Shifting into High Gear with Advanced Concepts

You can find each term by multiplying the previous term by 1 3 Here s another example to familiarize you with geometric formulas feel free to break it out at parties in the future An unwise gambler bets a dollar on the flip of a coin and loses Instead of paying up, he says, Double or nothing, meaning that he wants to flip the coin again; he ll pay two bucks if he loses, and his opponent gets nothing if the gambler wins Oops! He loses again, and again he says, Double or nothing! If he repeats this doubling-andlosing process 20 times, how much will he owe on the 21st try Using the formula gn = g1rn 1 (you know the first term the one dollar and the multiplier), you replace the first term, g1, with the number 1, r with 2 for the doubling, and n with 21: g 21 = 1 ^ 2h

21 - 1

= 1 ^ 2h = 1, 048, 576

The gambler will owe over one million dollars if he keeps going to 21 If he s unwilling to part with his first dollar, how s he going to deal with this number Maybe the gambler shouldn t start with such a big bet at the beginning What if he starts with a quarter rather than a dollar Using the same formula, gn = g1rn 1, the first term is 025, and the ratio is still 2 So, g21 = (025)220 = $262,144 Looks like he s still in big trouble And, in two more flips, letting n = 23, he reaches the starting point for the dollar bill

An alternate way to describe the terms of a sequence, in place of giving the general rule for the sequence, is to define the sequence recursively To do so, you identify the first term, or maybe a few of the initial terms, and describe how to find the rest of the terms by using the terms that come before them The recursive rule for arithmetic sequences is an = an 1 + d, and the recursive rule for geometric sequences is gn = rgn 1 Here s an example of a recursively defined sequence Let a1 = 6 and an = 2an 1 + 3 The formula says that, to find a term in the sequence, you look at the previous term (an 1), double it (2an 1), and add 3 The first term is 6, so the second term is 3 more than the double of 6, or 15 The next term is 3 more than the double of 15, or 33 Here are some of the terms of this sequence listed in order: {6, 15, 33, 69, 141, }

Sometimes, you have to list the numbers in a recursive sequence when given a rule, and, if you re really lucky, you get to create the rule yourself You ll see the create your own rule option if you get into discrete mathematics or computer programming just not here! You can also define sequences recursively by referring to more than one previous term For instance, assume that you let an = 3an 2 + an 1 This rule says that to find the nth term in the sequence (you choose the n you want the 5th term, the 50th term, and so on), you have to look at the two previous terms [the (n 2)nd and the (n 1)st terms], multiply the term two positions back by 3, 3(an 2), and then add the term one position back to the product, an 1 To start writing the terms of this sequence, you need to identify two consecutive terms For this particular sequence, you decide to let b1 = 4 and b2 = 1 (Okay, you didn t get to decide, I did I just picked random numbers, but, after they re chosen, they determine what s going to happen to the rest of the numbers in the sequence using the given rule) Here s how the terms go (Note: If you re looking for the sixth term, you need the [n 1]st term, which is the fifth term, and the [n 2]nd term, which is the fourth term): b 1 = 4, b 2 = -1 b n = 3b n - 2 + b n - 1

b 3 = 3b 1 + b 2 = 3 ^ 4 h + ^ -1h = 12 - 1 = 11 b 4 = 3b 2 + b 3 = 3 ^ -1h + 11 = - 3 + 11 = 8 b 5 = 3b 3 + b 4 = 3 ^11h + 8 = 33 + 8 = 41 b 6 = 3b 4 + b 5 = 3 ^ 8 h + 41 = 24 + 41 = 65

Recursively formed sequences use earlier terms in the sequences to form later terms The rules used to write these sequences aren t as handy as the rules that allow you to find the 50th or 100th term (like arithmetic and geometric sequences) without finding all the terms that come before But the recursive rule is sometimes easier to write For example, if you find out that your salary at a new job would be $20,000 this year, $25,000 next year, and that every year after it will be 80 percent of the salary from 2 years ago plus 40 percent of the salary from the previous year, would you be willing to sign the contract The rule reads: bn = 08bn 2 + 04bn 1 Using the first two terms and this rule, your salary for the first five years would be {20,000, 25,000, 26,000, 30,400, 32,960, }