2. 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 (www.mrgover.com)

3. 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

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

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

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

7. Example Find the common ratio: -2,4,-8… 10,20,30... 30,20,10...

8. Example Find the next 3 terms in the geometric sequence: 18,54,162... Steps: 1. Find the common ratio, r 2. Multiply each previous term by r to get next term

9. Try This Find the next 3 terms in the geometric sequence: 27,135,675... r =5 The next 3 terms are: 3375,16875,84375

10. Analysis Let a 1 =3 & r =4: Term No. Term Name Term 1 a 1 3 2 a 2 = a 1 r 1 12 3 a 3 = a 1 r 2 48 n a n =a 1 r n-1 ?

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

12. Example Find the 14th term in the sequence :

13. Try This Find the 12th term in the sequence :

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

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

16. Try This Form a sequence that has 2 geometric means between 128 & 54. 128,____,____,54 9672

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

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

19. Example Find the sum of the first 8 terms of the geometric series: 3-6+12+... Steps: 1. Find the value of r 2. Use the formula

20. Try This Find the sum of the first 6 terms of the geometric series: 2-6+18-54... -364

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

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

23. Assignment Pgs. 669/1-119 EOO