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Unformatted text preview: 1.7. SEQUENCES AND DIFFERENCE EQUATIONS 153 1.7 Sequences and Difference Equations Often, experimental measurements are collected at discrete intervals of time. For example, the number of elephants in wildlife park in Africa may be counted every year to ensure that poachers are not driving the population extinct in the near future. Blood may be drawn on a weekly basis from a patient infected with HIV and the number of CD4+ cells produced by patients immune system counted to monitor the progression of the patient towards full-blown AIDS. Data obtained in this regular fashion can be represented by a sequence of numbers over time. In this section, we describe the basic properties of such sequences and demonstrate that some sequences can be generated recursively using a relationship called a difference equation . These equations are formulated using a function from the natural numbers to the real numbers. Sequences We begin with the idea of a sequence, which is simply a succession of numbers that are listed according to a given prescription or rule. Specifically, if n is a natural number, the sequence whose n th terms is the number a n can be written as a 1 ,a 2 ,a 3 ,...,a n ,... The number a 1 is called the first term, a 2 the second term, ... , and a n the n th term. Sequence A sequence is a real-valued function whose domain is the set of natural numbers. When working with sequences, we alter the usual functional notation. For a function a from the natural to the real numbers we should write a (1) , a (2) , a (3) , ... , but for convenience we write a 1 , a 2 , a 3 ,... . The function a ( n ) is written a n and is called the general term . Example 1. Finding the sequence, given the general term Find the first five terms of the sequences whose general term is given. a. a n = n b. a n = sin n 2 c. a n = n 1+ n d. a n is the digit in the n th decimal place of the number . 2010 Schreiber, Smith & Getz 154 1.7. SEQUENCES AND DIFFERENCE EQUATIONS e. a 1 = 5 a n +1 = 2 a n for n 1 Solution. a. Since n is the general term, we have 1 , 2 , 3 , 4, and 5 for the first five terms. b. For n = 1 , sin 2 = 1; for n = 2 , sin 2 2 = 0; for n = 3 , sin 3 2 = 1; for n = 4 , sin 4 2 = 0; and for n = 5 , sin 5 2 = 1. c. Take the first five natural numbers (in order) to find: 1 1+1 = 1 2 , 2 1+2 = 2 3 , 3 1+3 = 3 4 , 4 1+4 = 4 5 , and 5 1+5 = 5 6 d. Since 3 . 141592 ; we see the first five terms of this sequence is: 1 , 4 , 1 , 5 , and 9. e. This is known as a recursive formula because after one (or more) given term(s), the subsequent terms are found in terms of the given term(s). For this example, the first term is given: a 1 = 5; for n = 2, we use a 2 = 2 a 1 = 2(5) = 10; for n = 3, a 3 = 2 a 2 = 2(10) = 20; for n = 4, a 4 = 2 a 3 = 2(20) = 40, and for n = 5, a 5 = 2(40) = 80. In summary, the first five terms of the sequence are 5 , 10 , 20 , 40 , 80....
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