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Mathematical Investigations IV Name Mathematical Investigations IV Iteration Forever SEQUENCES AND SERIES – A RECAP What the Notation Means The notation a nn1 means to write out the a’s in order as n runs from 1 to 5: a1, a2, a3, 5 5 a4, a5. The variable n is called an index. The notation an means to write out the a’s in order n1 5 as n runs from 1 to 5 and then add them. In other words: an a1 a2 a3 a4 a5 . If we n1 write out a set in order, it is called a sequence. If we do it with a sum, the result is called a series. These vocabulary words are not interchangeable! Expressing Sequences The sequence 3 5nn1 consists of the numbers 8, 3, –2, –7, –12, –17 since, as n 4 counts from –1 to 4, you get the numbers 8, 3, etc. in order. A succinct way to write this is to say an = 3 – 5n, which is called an explicit formula. It is called explicit because all you have to do to find the jth term is to drop j into the formula and it tells you explicitly what the value is. Other times, though, an may depend not only on n, but on previous values aj where j < n. Such formulas tell you at what number to start (what is the value for the smallest possible index value) and how to get from one term to the next. For example: 1 n0 an is the sequence 1, 1, 2, 6, ...(this is an alternative definition of n!). n an1 n 0 In order to find an you have to know the value of previous terms. Such sequences are called recursive. Note that this sequence started with the n = 0 term, and not n = 1. Some Special Kinds of Sequences Often we take a recursive sequence and put a bunch of different starting values in to see what happens. Starting values are called seeds and the path that the sequence takes is called an orbit. For some values of the seed the orbit doesn’t go anywhere—every term is the same. If an = f(an-1) is the recursive sequence, this will happen if an = an-1, and then this value will keep repeating forever. Such values are called fixed points and are found by solving x = f(x). A very handy kind of sequence that comes up quite often is one where each term is obtained from the previous one by adding some fixed number, i.e. an = an-1 + d for some fixed d. Thus if the sequence starts with a, it is followed by a + d, a + 2d, a + 3d, etc. Such sequences are called arithmetic. It is important to note that the first term has no d’s in it, the second has one, the third has two, the fourth has three, and so on, so that the nth has n – 1. In other words, we have the important formula an = a1 + (n - 1)d. The series obtained from adding all the terms of an arithmetic sequence is naturally called an arithmetic series. Another common sequence is called geometric. It is like an arithmetic sequence, except that instead of adding a fixed number each time, you multiply by a fixed number. Thus the Seq & Ser. 11.1 Rev. S07 Mathematical Investigations IV Name sequence begins a, ar, ar2, ar3, etc. Note that the first term has no r’s in it, the second has one, the third has two, and so on. The nth term will have n – 1 r’s in it, so an = arn-1. The series obtained by adding these up is called a geometric series. Finally, a sequence whose terms are reciprocals of an arithmetic sequence is called a harmonic sequence. So the terms of a harmonic sequence are 1/a, 1/(a + d), 1/(a + 2d), .... The nth term can be found by taking the reciprocal of the nth term of the corresponding arithmetic sequence, so an = 1/(a + (n - 1)d). Handy Formulas Arithmetic first term = a common difference = d nth term = a + (n – 1)d sum of first n terms: n Geometric first term = a common ratio = r nth term = arn-1 sum of first n terms: n n a k 1d 2 2a n 1d k 1 (Note that since the nth term is a + (n – 1)d, this can be rewritten n(a1 + an)/2, where a1 = a is the first term, and an is the last term) Infinite arithmetic series have no value (they diverge) ar k 1 k 1 1 r n a 1 r (Note that any time you have a geometric series, all you need to do is plug in the first term, a, and the number of terms, n, in this formula to get the sum.) If |r| < 1, the infinite geometric series has the value a /(1 – r). This can be found by seeing that rn 0 in the above as n gets very large. To Emphasize Anytime you have an arithmetic series, all you need to know is the first term a, the common difference d (or the last term, an), and the number of terms n, to determine the sum, n(2a + (n – 1)d)/2 or n(a1 + an) / 2. Similarly for a geometric series, all you need is the first term a, the common ratio r, and the number of terms n to determine the sum, a(1 – rn) / (1 – r). Or for an infinite geometric series with |r| < 1, all you need is a and r to determine the sum, a / (1 – r). Some Special Sums n k 1 2 3 n k 1 n k k 1 n(n 1) 2 n k 2 12 22 32 k 1 n2 n(n 1)(2n 1) 6 2 3 3 3 3 1 2 3 n n(n 1) 2 n k k 1 2 3 Seq & Ser. 11.2 Rev. S07