2Geometric SequenceThe ratio of a term to it’s 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).
3Example: Decide whether each sequence is geometric. 4,-8,16,-32,…-8/4=-216/-8=-2-32/16=-2Geometric (common ratio is -2)3,9,-27,-81,243,…9/3=3-27/9=-3-81/-27=3243/-81=-3Not geometric
4Rule for a Geometric Sequence an=a1rn-1Example: 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 = .4an=5(.4)n-1a8=5(.4)8-1a8=5(.4)7a8=5( )a8=
5Example: One term of a geometric sequence is a4=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.If a4=3, then when n=4, an=3.Use an=a1rn-13=a1(3)4-13=a1(3)33=a1(27)1/9=a1an=a1rn-1an=(1/9)(3)n-1To graph, graph the points of the form (n,an).Such as, (1,1/9), (2,1/3), (3,1), (4,3),…
6Example: Two terms of a geometric sequence are a2=-4 and a6=-1024 Example: Two terms of a geometric sequence are a2=-4 and a6= Write a rule for the nth term.Write 2 equations, one for each given term.a2=a1r2-1 OR -4=a1ra6=a1r6-1 OR =a1r5Use these 2 equations & substitution to solve for a1 & r.-4/r=a1-1024=(-4/r)r5-1024=-4r4256=r44=r & -4=rIf r=4, then a1=-1.an=(-1)(4)n-1If r=-4, then a1=1.an=(1)(-4)n-1an=(-4)n-1Both Work!
7Formula for the Sum of a Finite Geometric Series n = # of termsa1 = 1st termr = common ratio
8Example: Consider the geometric series 4+2+1+½+… . Find the sum of the first 10 terms.Find n such that Sn=31/4.