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11.3 Geometric Sequences & Series

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

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Example: Decide whether each sequence is geometric. 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

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

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

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Example: Two terms of a geometric sequence are a 2 =-4 and a 6 =-1024. Write a rule for the nth term. Write 2 equations, one for each given term. a 2 =a 1 r 2-1 OR -4=a 1 r a 6 =a 1 r 6-1 OR -1024=a 1 r 5 Use these 2 equations & substitution to solve for a 1 & r. -4 / r =a 1 -1024=( -4 / r )r 5 -1024=-4r 4 256=r 4 4=r & -4=r If r=4, then a 1 =-1. a n =(-1)(4) n-1 If r=-4, then a 1 =1. a n =(1)(-4) n-1 a n =(-4) n-1 Both Work!

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

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

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log 2 32=n

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Assignment

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