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Geometric Sequences and Series

Section 9-3. Geometric Sequences and Series. Objectives. Recognize, write, and find nth terms of geometric sequences Find the nth partial sums of geometric sequences Find the sum of an infinite geometric sequence. Definition of a Geometric Sequence.

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Geometric Sequences and Series

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  1. Section 9-3 Geometric Sequences and Series

  2. Objectives • Recognize, write, and find nth terms of geometric sequences • Find the nth partial sums of geometric sequences • Find the sum of an infinite geometric sequence

  3. Definition of a Geometric Sequence • A geometric sequence is a sequence in which each term after the first is obtained by multiplying the preceding term by a fixed nonzero constant. The amount by which we multiply each time is called the common ratio of the sequence.

  4. Definition of Sequence An infinite sequence is a function whose domain is the set of positive integers. a1, a2, a3, a4, . . . , an, . . . terms The first three terms of the sequence an = 2n2 are a1= 2(1)2 = 2 finitesequence a2= 2(2)2 = 8 a3= 2(3)2 = 18.

  5. Definition of Geometric Sequence Asequence is geometric if the ratios of consecutive terms are the same. 2, 8, 32, 128, 512, . . . geometric sequence The common ratio, r, is 4.

  6. General Term of a Geometric Sequence • The nth term (the general term) of a geometric sequence with the first term a1 and common ratio r is • an = a1 r n-1

  7. The nth Term of a Geometric Sequence a2 = 15(5) a3 = 15(52) a4 = 15(53) The nth term of a geometric sequence has the form an = a1rn - 1 where r is the common ratio of consecutive terms of the sequence. a1 = 15 15, 75, 375, 1875, . . . The nth term is: an = 15(5)n-1.

  8. Example: Finding the nth Term Example: Find the 9th term of the geometric sequence 7, 21, 63, . . . a1 = 7 an = a1rn – 1 = 7(3)n – 1 a9 = 7(3)9 – 1 = 7(3)8 = 7(6561) = 45,927 The 9th term is 45,927.

  9. The Sum of the First n Terms of a Geometric Sequence The sum, Sn, of the first n terms of a geometric sequence is given by in which a1 is the first term and r is the common ratio.

  10. Example • Find the sum of the first 12 terms of the geometric sequence: 4, -12, 36, -108, ... Solution:

  11. Definition of Summation Notation The sum of the first n terms of a sequence is represented by summation notation. upper limit of summation lower limit of summation index of summation

  12. The Sum of a Finite Geometric Sequence The sum of a finite geometric sequence is given by 5 + 10 + 20 + 40 + 80 + 160 + 320 + 640 = ? n = 8 a1 = 5

  13. If -1<r<1, then the sum of the infinite geometric series a1+a1r+a1r2+a1r3+… in which a1 is the first term and r s the common ration is given by The Sum of an Infinite Geometric Series If |r|>1, the infinite series does not have a sum.

  14. Example: Sum of Infinite Geometric Series The sum of the series is Example: Find the sum of

  15. Homework • WS 13-5

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