12.3 Geometric Sequences and Series ©2001 by R. Villar All Rights Reserved

Arithmetic Sequences ADD To get next term Geometric Sequences MULTIPLY To get next term Arithmetic Series Sum of Terms Geometric Series Sum of Terms

Geometric Sequences and Series Geometric Sequence: sequence whose consecutive terms have a common ratio. 1, 3, 9, 27, 81, 243,... The terms have a common ratio of 3. The common ratio is the number r =. Example Is the sequence geometric? 4, 6, 9, 13.5, 20.25, … Yes, the common ratio is 1.5 To find any term in a geometric sequence, use the formula a n = a 1 r n–1 where r is the common ratio.

Example: Find the common ration of a geometric sequence if the first term is 8 and the 11 th term is 1/128. a n = a 1 r n–1 a 1 = 9 r = 1.2 a 9 = a 12 = Example: Find the twelfth term of the geometric sequence whose first term is 9 and whose common ratio is 1.2. a n = a 1 r n–1 a 1 = 8 a 11 = 1/128 1/128 = 8 r 10 1/1024 = r 10 ½ = r

x 5 NA

x 9

Find two geometric means between –2 and 54 -2, ____, ____, NA x The two geometric means are 6 and -18, since –2, 6, -18, 54 forms an geometric sequence

Vocabulary of Sequences (Universal) Which can be simplified to: To find the sum of a geometric series, we can use summation notation.

Example: Evaluate the sum of: Convert this to =

1/2 7 x

1, 4, 7, 10, 13, …. Infinite Arithmetic No Sum 3, 7, 11, …, 51 Finite Arithmetic 1, 2, 4, …, 64 Finite Geometric 1, 2, 4, 8, …Infinite Geometric r > 1 r < -1 No Sum Infinite Geometric -1 < r < 1

Find the sum, if possible:

The Bouncing Ball Problem – Version A A ball is dropped from a height of 50 feet. It rebounds 4/5 of it’s height, and continues this pattern until it stops. How far does the ball travel? / /5

The Bouncing Ball Problem – Version B A ball is thrown 100 feet into the air. It rebounds 3/4 of it’s height, and continues this pattern until it stops. How far does the ball travel? / /4