11.3 Geometric Sequences & Series By: L. Keali’i Alicea

Geometric Sequence 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 number that you multiply by is called the common ratio (r).

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

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

Example: 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-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: Two terms of a geometric sequence are a2=-4 and a6=-1024 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!

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

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

log232=n

Assignment 11.3 A (all)