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Chapter 9.3 Geometric Sequences and Series

Chapter 9.3 Geometric Sequences and Series. Introduction to Geometric Sequences. As opposed to an arithmetic sequence that has a common difference (d) , geometric sequences have a common ratio (r)

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Chapter 9.3 Geometric Sequences and Series

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  1. Chapter 9.3 Geometric Sequences and Series

  2. Introduction to Geometric Sequences • As opposed to an arithmetic sequence that has a common difference (d), geometric sequences have a common ratio (r) • What is a ratio? Given the below geometric sequence, how could we determine the ratio of the terms? • Ex: 5, 15, 45, 135 • If we take we can see that the common ratio is 3

  3. Definition of Geometric Sequences • A sequence is geometric if the ratios of consecutive terms are the same. • a1, a2, a3, a4, . . ., an is geometric if there is a common ratio, r , such that

  4. Examples of Geometric Sequences • The sequence whose nth term is 2n is geometric. For this sequence, what is the common ratio? • The sequence whose nth term is 4(3n) is geometric. For this sequence, what is the common ratio? • Find the common ratio of this sequence. • 3, 12, 48, 192

  5. Finding the nth Term of a Geometric Sequence • The nth term of a geometric sequence has the form • where r is the common ratio of consecutive terms.

  6. Find the Terms of the Sequence • Write the first five terms of the geometric sequence whose first term is a1 = 3 and whose common ratio is r = 2. • You Try: a1 = 6, r = 2

  7. Finding a Term of a Geometric Sequence • Find the 15th term of the geometric sequence whose first term is 20 and whose common ratio is 1.05 • Find the 12th term of the geometric sequence5, 15, 45 • You Try: a1 = 4, r = ½, n= 10

  8. Finding a Term of a Geometric Sequence • The fourth term of a geometric sequence is 125, and the 10th term is . Find the 14th term. • You Try: Find the indicated term.6th term, a4 = -18, a7 = 2/3

  9. Sums of a Finite Geometric Sequences • The sum of the finite geometric sequence • with common ratio r ≠ 1 is given by

  10. Finding the Sum of a Finite Geometric Sequence • Find the sum • Find the sum:

  11. Increasing Annuity • A deposit of $50 is made on the first day of each month in a savings account that pays 6% compounded monthly. What is the balance at the end of 2 years? • Recall that the formula for compound interest is

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