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8.3 Geometric Sequences and Series Objectives: -Students will recognize, write, and find the nth terms of geometric sequences. -Students will find the nth partial sums of geometric sequences. -Students will find sums of infinite geometric series. -Students will use geometric sequences to model and solve real-life problems.

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Definition of Geometric Sequence A sequence is geometric if the ratios of consecutive terms are the same. a 1, a 2, a 3, a 4, …, a n, … is geometric if there is a number r such that r is the common ratio of the geometric sequence

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Ex 1) Determine whether the sequence is geometric. If it is, find the common ratio. a) 6, 18, 30, 42, … b) 1, -½, ¼, -⅛, …

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The nth Term of a Geometric Sequence The nth term of a geometric sequence has the form a n = a 1 · r n – 1 where r is the common ratio.

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Ex 2) Write the first 5 terms of the geometric sequence. Find the common ratio and write the nth term of the sequence as a function of n. a 1 = 64 and a k+1 = ½ a k

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Ex 3) Find the nth term of the geometric sequence. Use the table feature of your calculator to verify your answer. a 1 =4, r = ½, n = 10

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Ex 4) Find the nth term of the geometric sequence. Use the table feature of your calculator to verify your answer. a 2 = -18, a 5 =2/3, n = 6

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The Sum of an Infinite Geometric Sequence If │r│ 1, the series does not have a sum. The Sum of a Finite Geometric Sequence

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Ex 5) Find the sum. Use a calculator to verify.

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Ex 6) Find the sum. Use a calculator to verify.

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Ex 7) Use summation notation to write the sum. 5 + 15 + 45 + … + 3645

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Ex 8) Find the sum of the infinite geometric series, if possible. If not possible, explain why.

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Ex 9) Find the sum of the infinite geometric series, if possible. If not possible, explain why.

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Ex 10) Find the rational number representation of the repeating decimal. a) b)

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