11.3 Geometric Sequences & Series

What is a geometric sequence? What is the rule for a geometric sequence? How do you find the nth term given 2 terms?

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).

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

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( ) a 8 =

What is a geometric sequence? A sequence where the ratio of any term to the previous term is constant. What is the rule for a geometric sequence? a n =a 1 r n-1 How do you find the nth term given 2 terms? Write two equations with two unknowns and solve by substitution.

11.3 Assignment, Day 1 Page 670, all

Geometric Sequences and Series day 2 What is the formula for finding the sum of an finite geometric series?

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),…

Example: Two terms of a geometric sequence are a 2 =-4 and a 6 = 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 =( -4 / r )r =-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!

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

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

log 2 32=n

What is the formula for finding the sum of an finite geometric series?

11.3 Assignment, Day 2 p odd. Skip 63