1063 11.3 Geometric Sequences and Series Sum of the First n Terms of a Geometric Sequence If a geometric sequence has first term a1 and common ratio r, then the sum Sn of the first n terms is given by the following. Sn = a111 −r n2 1 −r , where r ≠1 EXAMPLE 5 Finding the Sum of the First n Terms At the beginning of this section, we found that an employee agreed to work for the following salary: $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 altogether in 20 days? SOLUTION We must find the total amount earned in 20 days with daily wages of 1, 2, 4, 8, . . . cents. Sn = a111 - r n2 1 - r Formula for Sn S20 = 111 - 2202 1 - 2 Let n = 20, a1 = 1, and r = 2. S20 = 1 - 1,048,576 -1 Evaluate 220. Subtract in the denominator. S20 = 1,048,575 cents, or $10,485.75 Evaluate. S Now Try Exercise 11(b). Geometric Series A geometric series is the sum of the terms of a geometric sequence. For example, a scientist might want to know the total number of insects in four generations of the population discussed in Example 4. This population would equal a1 + a2 + a3 + a4. 100 + 10011.52 + 10011.522 + 10011.523 = 812.5 ≈813 insects To find a formula for the sum of the first n terms of a geometric sequence Sn, first write the sum as Sn = a1 + a2 + a3 + g+ an or Sn = a1 + a1r + a1r 2 + g+ a 1r n-1. (1) If r = 1, then Sn = na1, which is a correct formula for this case. If r ≠1, then multiply both sides of equation (1) by r to obtain rSn = a1r + a1r 2 + a 1r 3 + g+ a 1r n. (2) Now subtract equation (2) from equation (1), and solve for Sn. Sn = a1 + a1r + a1r 2 + g+ a 1r n-1 (1) rSn = a1r + a1r 2 + g+ a 1r n-1 + a 1r n (2) Sn - rSn = a1 - a1r n Subtract. Sn11 - r2 = a111 - r n2 Factor. Sn = a111 - r n2 1 - r , where r ≠1 Divide by 1 - r.
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