9.3: Geometric Sequences and Series Objectives: Find the n th term and geometric mean Find the sum of n terms ©2002 Roy L. Gover (

Review What is the common difference in the following sequences: 1,5,9… 12,2,-8 2,4,8... The last sequence does not have a common difference. It has a common________. ratio

Definition A geometric sequence is a sequence in which each term after the first is the previous term multiplied by a common ratio.

Example Identify the following as either arithmetic or geometric sequences: 3,7,11… -2,4,-8… 3,-2,-7

Defintion The common ratio of a geometric sequence is found by dividing any term by the preceding term.

Example Find the common ratio: -2,4,-8… 10,20, ,20,10...

Example Find the next 3 terms in the geometric sequence: 18,54, Steps: 1. Find the common ratio, r 2. Multiply each previous term by r to get next term

Try This Find the next 3 terms in the geometric sequence: 27,135, r =5 The next 3 terms are: 3375,16875,84375

Analysis Let a 1 =3 & r =4: Term No. Term Name Term 1 a a 2 = a 1 r a 3 = a 1 r 2 48 n a n =a 1 r n-1 ?

Definition The nth term of a geometric sequence with first term a 1 and common ratio r is given by the following formula:

Example Find the 14th term in the sequence :

Try This Find the 12th term in the sequence :

Definition The terms between any 2 nonconsecutive terms of a geometric sequence are called geometric means.

Example Form a sequence that has 2 geometric means between 136 & ,____,____,459 If we only knew the value of r...

Try This Form a sequence that has 2 geometric means between 128 & ,____,____,

Definition A geometric series is the sum of the terms of a geometric sequence. Sequence: 2,6,18,… Series:

Definition The sum of the first n terms of a geometric series is:

Example Find the sum of the first 8 terms of the geometric series: Steps: 1. Find the value of r 2. Use the formula

Try This Find the sum of the first 6 terms of the geometric series:

Try This Find the sum of the first 6 terms of the geometric series:

Lesson Close In the formula for the sum of a geometric series: Why can a 1 not be 0?

Assignment Pgs. 669/1-119 EOO