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9.3 Geometric Sequences and Series

9.3 Geometric Sequences and Series. Objective. To find specified terms and the common ratio in a geometric sequence. To find the partial sum of a geometric series. Geometric Sequences. Consecutive terms of a geometric sequence have a common ratio. Definition of a Geometric Sequence.

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9.3 Geometric Sequences and Series

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  1. 9.3 Geometric Sequences and Series

  2. Objective • To find specified terms and the common ratio in a geometric sequence. • To find the partial sum of a geometric series

  3. Geometric Sequences • Consecutive terms of a geometric sequence have a common ratio.

  4. Definition of a Geometric Sequence • A sequence is geometric if the ratios of consecutive terms are the same. • The number r is the common ratio of the sequence.

  5. Example 1Examples of Geometric Sequences • a). The sequence whose nth term is • b). The sequence whose nth term is • C) The sequence whose nth term is

  6. Notice that each of the geometric sequences has an nth term that is of the form where the common ratio is r. • A geometric sequence may be thought of as an exponential function whose domain is the set of natural numbers.

  7. 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 of the sequence.

  8. So, every geometric sequence can be written in the following form,

  9. If you know the nth term of a geometric sequence, you can find the (n+1)th term by multiplying by r. that is

  10. Example 2Finding the Terms of a Geometric Sequence Write the first five terms of the geometric sequence whose first term is and whose common ratio is r = 2. 3, 6, 12, 24, 48

  11. Example 3Finding a Term of a Geometric Sequence • Find the 15th term of the geometric sequence whose first term is 20 and whose common ration is 1.05.

  12. Example 4Finding a Term of a Geometric Sequence • Find the 12th term of the geometric sequence 5, 15, 45, . . .

  13. If you know any two terms of a geometric sequence, you can use that information to find a formula for the nth term of the sequence.

  14. Example 5Finding a Term of a Geometric Sequence • The fourth term of a geometric sequence is 125, and the 10th term is 125/64. Find the 14th term. (assume that the terms of the sequence are positive.)

  15. The Sum of a Finite Geometric Sequence • The sum of the geometric sequence with common ratio is given by

  16. Example 6Finding the Sum of a Finite Geometric Sequence • Find the sum

  17. When using the formula for the sum of a finite geometric sequence, be careful to check that the index begins at . If the index begins at , you must adjust the formula for the th partial sum.

  18. These are not the same, be careful of the indices

  19. Geometric Series • The summation of the terms of an infinite geometric sequence is called an infinite geometric series or geometric series.

  20. The sum of an Infinite Geometric Series • If the infinite geometric series has the sum

  21. Example 7Finding the Sums of an Infinite Geometric Series • Find the sums. • a) • b)

  22. ApplicationsCompound Interest • 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 of this annuity at the end of 2 years?

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