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 =
n 2. S = a1 − an+1
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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