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Chapter 8 Sec 3 Geometric Sequences and Series

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2 of 15 Pre Calculus Ch 8.3 Essential Question How do you find terms and sums of geometric sequences? Key Vocabulary: Geometric Sequence Common ratio

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3 of 15 Pre Calculus Ch 8.3 Definition of a Geometric Sequence A sequence is geometric if the ratio of consecutive terms are the same. So, the sequence is geometric if there is a number r such that The number r is the common ratio of the sequence.

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4 of 15 Pre Calculus Ch 8.3 Begin with n = 1. The common ratio is 2. The common ratio is 2. Example 1 a. The sequence whose n th term is 2 n is geometric. Begin with n = 1. The common ratio is 3. b. The sequence whose n th term is 4(3 n ) is geometric.. c. The sequence whose n th term is n 2 is NOT geometric.

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5 of 15 Pre Calculus Ch 8.3 The n th term of a Geometric Sequence The n th term of an geometric sequence has the form Where r is the common ratio between consecutive terms of the sequence and every geometric sequence can be written in the following form. If you know the n th term, you can find the (n + 1) th term by multiplying by r.

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6 of 15 Pre Calculus Ch 8.3 Example 2 Write the first five terms of the geometric sequence whose common ratio is r = 2 and whose first term is a 1 = 3. a 1 = 3 a 2 = 3(2 1 )=6 a 3 = 3(2 2 )=12 a 4 = 3(2 3 )=24 a 5 = 3(2 4 )=48

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7 of 15 Pre Calculus Ch 8.3 Example 3 The 15 th term of the geometric sequence where a 1 = 20 and r = 1.05. a n = a 1 r n – 1 a 15 = 20 (1.05) 15 – 1 a 15 = 20 (1.05) 14

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8 of 15 Pre Calculus Ch 8.3 Example 4 Find a formula for the n th term of the following geometric sequence and what is the ninth term of the sequence? 5, 15, 45, …

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9 of 15 Pre Calculus Ch 8.3 Example 5 The 4 th term of a geometric sequence is 125 and the 10 th term is 125/64. Find the 14 th term. (Assume that the terms are all positive.)

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10 of 15 Pre Calculus Ch 8.3 The Sum of a Finite Geometric Sequence The Sum of a Finite Geometric Sequence with common ratio r ≠ 1 is given by Find the sum a 1 = 4(0.3), r = 0.3, and n = 12… When using this formula, be careful to check index begins at i = 1. If not then you’ll need to adjust sum.

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11 of 15 Pre Calculus Ch 8.3 Geometric Series The Sum of a Infinite Geometric Series If | r | < 1, then the infinite geometric series has the sum Note: if | r | ≥ 1 the series has no sum. The sum of an infinite geometric sequence is called infinite geometric series or geometric series. If r has the property that | r | < 1, then it can be shown r n become arbitrarily close to zero as n increases without bound.

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12 of 15 Pre Calculus Ch 8.3 Example 7 Find the sixth partial sum of the series, then find the sum of the series. a 1 = 4 and r = 0.6

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13 of 15 Pre Calculus Ch 8.3 Example 8 Find the sum 3 + 0.3 + 0.03 + 0.003 + … a 1 = 3 and Compounded Interest Formulas Compounded Daily, Quarterly,… Compounded Continuously (only)

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14 of 15 Pre Calculus Ch 8.3 Essential Question How do you find terms and sums of geometric sequences?

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15 of 15 Pre Calculus Ch 8.3 Daily Assignment Chapter 8 Section 3 Text Book Pg 607 – 608 All ODD OF: #1 – 21; #33 – 35; #55 – 65; #73 – 75; #83 – 85 Read Section 8.5 Show all work for credit.

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