summation

# The enumeration of terms with ellipses becomes

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Unformatted text preview: . The enumeration of terms with ellipses becomes tedious. 18 Geometric Series If a and r ￿= 0 are real numbers, then ￿ n+1 n ar −a ￿ if r ￿= 1 j r −1 ar = (n + 1)a if r = 1 j =0 Proof: The case r = 1 holds, since arj = a for each of the n + 1 terms of the sum. The case r ￿= 1 holds, since n n ￿ ￿ ￿n (r − 1) j =0 arj = arj +1 − arj = j =0 n+1 ￿ j =0 arj − j =1 n+1 n ￿ arj j =0 = ar −a and dividing by (r − 1) yields the claim. 19 Sum of First n Positive Integers For all n ≥ 1, we have n ￿ k = n(n + 1)/2 k=1 We prove this by induction. Basis step: For n = 1, we have 1 ￿ k = 1 = 1(1 + 1)/2. k=1 20 Sum of the First n Positive Integers Induction Hypothesis: We assume that the claim holds for n − 1. Induction Step: Assuming the Induction Hypothesis, we will show that the claim holds for n. ￿n k=1 k = = = = ￿ n− 1 n + k=1 k 2n/2 + (n − 1)n/2 by Induction Hypothesis 2n+n2 −n 2 n(n+1) 2 Therefore, the claim follows by induction on n. 21 Sum of Fibonacci Nu...
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