11.3 Geometric Sequences & Series

11.3 Geometric Sequences & Series

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).

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.

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

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.

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

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

log232=n

Assignment

11.3 Geometric Sequences & Series