Stitz-Zeager_College_Algebra_e-book

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Unformatted text preview: 1−r In the case when r = 1, we get the formula S = a + a + ... + a = na n times Our results are summarized below. Equation 9.2. Sums of Arithmetic and Geometric Sequences: • The sum S of the first n terms of an arithmetic sequence ak = a + (k − 1)d for k ≥ 1 is n S= ak = n k=1 a1 + an 2 = n (2a + (n − 1)d) 2 • The sum S of the first n terms of a geometric sequence ak = ark−1 for k ≥ 1 is n 1. S = ak = k=1 n 2. S = a1 − an+1 =a 1−r n ak = k=1 1 − rn , if r = 1. 1−r a = na, if r = 1. k=1 9.2 Summation Notation 567 While we have made an honest effort to derive the formulas in Equation 9.2, formal proofs require the machinery in Section 9.3. An application of the arithmetic sum formula which proves useful in Calculus results in formula for the sum of the first n natural numbers. The natural numbers themselves are a sequence4 1, 2, 3, . . . which is arithmetic with a = d = 1. Applying Equation 9.2, 1 + 2 + 3 + ... + n = n(n + 1) 2 So, for example, the sum of the first 100 natural numbers is 100(101) 2 = 5050.5 An important application of the geometric sum formul...
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