1061 11.3 Geometric Sequences and Series 11.3 Geometric Sequences and Series ■ Geometric Sequences ■ Geometric Series ■ Infinite Geometric Series ■ Annuities n thTerm of a Geometric Sequence In a geometric sequence with first term a1 and common ratio r, the nth term an is given by the following. an =a1 r n−1 EXAMPLE 1 Finding the n thTerm of a Geometric Sequence Use the formula for the nth term of a geometric sequence to answer the first question posed at the beginning of this section: How much will be earned on day 20 if daily wages follow the sequence 1, 2, 4, 8, 16, . . . cents? SOLUTION an = a1r n-1 Formula for a n a20 = 112220-1 Let n = 20, a 1 = 1, and r = 2. a20 = 524,288 cents, or $5242.88 Evaluate. S Now Try Exercise 11(a). Geometric Sequences Suppose an employee agrees to work for $0.01 the first day, $0.02 the second day, $0.04 the third day, $0.08 the fourth day, and so on, with wages doubling each day. How much is earned on day 20? How much is earned altogether in 20 days? These questions will be answered in this section. A geometric sequence (or geometric progression) is a sequence in which each term after the first is obtained by multiplying the preceding term by a fixed nonzero real number, called the common ratio. The sequence discussed above, 1, 2, 4, 8, 16, c , In cents is a geometric sequence in which the first term is 1 and the common ratio is 2. Notice that if we divide any term after the first term by the preceding term, we obtain the common ratio r = 2. a2 a1 = 2 1 = 2; a3 a2 = 4 2 = 2; a4 a3 = 8 4 = 2; a5 a4 = 16 8 = 2 If the common ratio of a geometric sequence is r, then r = an+1 an , Common ratio r for every positive integer n. Therefore, we find the common ratio by choosing any term after the first and dividing it by the preceding term. In the geometric sequence 2, 8, 32, 128, . . . , r = 4. Notice that 8 = 2 # 4 32 = 8 # 4 = 12 # 42 # 4 = 2 # 42 128 = 32 # 4 = 12 # 422 # 4 = 2 # 43. To generalize this, assume that a geometric sequence has first term a1 and common ratio r. The second term is a2 = a1r, the third is a3 = a2r = 1a1r2r = a1r 2, and so on. Following this pattern, the nth term is an = a1r n-1.
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