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9.3 Geometric Sequences & Series

9.3 Geometric Sequences & Series. Geometric Sequence. The ratio of a term to its previous term is constant. This means you multiply by the same number to get each term. This number that you multiply by is called the common ratio (r). 4,-8,16,-32,… -8 / 4 =-2 16 / -8 =-2 -32 / 16 =-2

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9.3 Geometric Sequences & Series

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  1. 9.3 Geometric Sequences & Series

  2. Geometric Sequence • The ratio of a term to its previous term is constant. • This means you multiply by the same number to get each term. • This number that you multiply by is called the common ratio (r).

  3. 4,-8,16,-32,… -8/4=-2 16/-8=-2 -32/16=-2 Geometric (common ratio is -2) 3,9,-27,-81,243,… 9/3=3 -27/9=-3 -81/-27=3 243/-81=-3 Not geometric Example: Decide whether each sequence is geometric.

  4. Rule for a Geometric Sequence an=a1rn-1 • Example: Write a rule for the nth term of the sequence 5, 2, 0.8, 0.32,… . Then find a8. • First, find r. • r= 2/5 = .4 • an=5(.4)n-1 a8=5(.4)8-1 a8=5(.4)7 a8=5(.0016384) a8=.008192

  5. If a4=3, then when n=4, an=3. Use an=a1rn-1 3=a1(3)4-1 3=a1(3)3 3=a1(27) 1/9=a1 an=a1rn-1 an=(1/9)(3)n-1 To graph, graph the points of the form (n,an). Such as, (1,1/9), (2,1/3), (3,1), (4,3),… Example: One term of a geometric sequence is a4=3. The common ratio is r=3. Write a rule for the nth term. Then graph the sequence.

  6. Example: Two terms of a geometric sequence are a2=-4 and a6=-1024. Write a rule for the nth term. • Write 2 equations, one for each given term. a2=a1r2-1 OR -4=a1r a6=a1r6-1 OR -1024=a1r5 • Use these 2 equations & substitution to solve for a1 & r. -4/r=a1 -1024=(-4/r)r5 -1024=-4r4 256=r4 4=r & -4=r If r=4, then a1=-1. an=(-1)(4)n-1 If r=-4, then a1=1. an=(1)(-4)n-1 an=(-4)n-1 Both Work!

  7. Formula for the Sum of a Finite Geometric Series n = # of terms a1 = 1st term r = common ratio

  8. Find the sum of the first 10 terms. Find n such that Sn=31/4. Example: Consider the geometric series 4+2+1+½+… .

  9. log232=n

  10. Sum of an infinte geometric sum:S= a1 / 1-r

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