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Geometric Sequences and Series Unit 10.3

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Practical Application “The company has been growing geometrically”

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Purpose 1.Calculate for a geometric sequence 2.Calculate for the nth term 3.Compute for geometric means 4.Calculate for sums of geometric series 5.Calculate for sum in sigma notation

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Calculate for geometric sequence 1.4, 11, 30.25 What are the next 4 in the series? Find common Ratio - Second term divided by first term 11 /4 = 2.75: common ratio is 2.75 30.25 * 2.75 = 83.19 83.19 * 2.75 = 228.77 228.77 * 2.75 = 629.11

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Page 615 Problems 1 - 6 Find the common ratio and the next 3 terms 1.Common ratio = -2 2, -4, 8 2.Common ratio = -.75 - 27/128 81/512 -243/2,048 3.Common ratio = 1.5 1.69, 2.53, 3.8 4.Common ratio = 2.5 125, 312.5, 781.25 5.Common ratio = 5 250x, 1250x, 6250x 6.Common ratio = ¼ x, 1/4x, 1/16x

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Formula a n = a 1 r n – 1 a n = final term in sequence a 1 = first term in sequence r = ratio n = term in sequence

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Example 1.Find the 9 th term of the geometric sequence 4, 14, 49 1.Find the ratio 14/4 = 7/2 or 3.5 2.a n = 4(3.5) 8 3.a n = 4(22518.75) 4.a n = 90,075.8

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Problems Page 615 Problems 20 - 24

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Geometric Means Write a sequence that has two geometric means between 480 and -7.5 Step 1. 480, a, b, -7.5 (4 terms) Step 2. a 4 = a 1 r n-1 -7.5 = 480r 4-1 -7.5 = 480r 3 -7.5/480 = r 3 -1/64 = r 3 -1/4 = r a 2 =-120 =(480*-.25) a 3 =30 = (-120*-.25)

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Problems Unit 10.3 Page 615 Problems 32 - 36

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Geometric Series Geometric Sequence 2, 4, 8, 16…. Geometric Series is the sum of the terms of a geometric sequence 2 + 4 + 8 + 16… Formulas S n = a 1 (1 – r n )/(1 – r) S n = a 1 – a n r (1 – r)

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Explanation of Formulas a) S n = a 1 (1 – r n )/(1 – r) b) S n = a 1 – a n r nth partial sum (1 – r)

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Examples Page 612 Guided practice 6a Find the first 11 terms 7 + (-24.5) + 85.75 a 1 = 7 r = -24.5/7 = -3.5 s 11 = 7(1 – (3.5) 11 /(1 – (-3.5)) s 11 = 1,501,877.34

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Examples Page 612 Guided practice 6b Find the sum of the first n terms a 1 = -8 a n = 131,072 r = -4 S n = a 1 – a n r 1 – r = -8 - (131,072)(-4)= 104,856 1 – (-4)

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Geometric Sum in Sigma Notation 7 ∑ 3(5) n – 1 n = 2 See board

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Sum of an infinite Geometric Series S = a 1 1 – r Problem: Find the sum of 10, -5, 2.5 r = -.5, a 1 = 10 S = 10/(1 – (-.5) = 6.67

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Problems Page 605-6 Problems 40 – 46, 48 – 52, 56,57

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