2. What you really need to know! A geometric sequence is a sequence in which the quotient of any two consecutive terms, called the common ratio, is the same. In the sequence 1, 4, 16, 64, 256,.., the common ratio is 4.

3. G e o m e t r i c S e r i e s

4. 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).

5. Example: Decide whether each sequence is geometric. 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

6. Rule for a Geometric Sequence u n =u 1 r n-1 Example: Write a rule for the n th term of the sequence 5, 2, 0.8, 0.32,…. Then find u 8. First, find r.First, find r. r= 2 / 5 = 0.4r= 2 / 5 = 0.4 u n =5(0.4) n-1u n =5(0.4) n-1 u 8 =5(0.4) 8-1 u 8 =5(0.4) 7 u 8 =5(0.0016384) u 8 =0.008192

7. One term of a geometric sequence is u 4 = 3. The common ratio is r = 3. Write a rule for the nth term. Then graph the sequence. If u 4 =3, then when n=4, u n =3.If u 4 =3, then when n=4, u n =3. Use u n =u 1 r n-1Use u n =u 1 r n-1 3=u 1 (3) 4-1 3=u 1 (3) 3 3=u 1 (27) 1 / 9 =a 1 u n =u 1 r n-1u n =u 1 r n-1 u n =( 1 / 9 )(3) n-1 To graph, graph the points of the form (n,u n ).To graph, graph the points of the form (n,u 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),…

8. Please find the 15 th term 5, 10, 20, 40 So, geometric sequence with u 1 = 5 r = 2

9. Compound Interest

10. 8 - 9 100 110 121 1000 1210 1331 1100 100 110 Time(Years) 0 1234 Amount $1000 110 Interest 100 Interest 133.1 Compounding Period Interest 121 Compound Interest - Future Value

11. COMPOUND INTEREST FORMULA FV is the Future Value in t years P is the Present Value amount started with r is the annual interest rate n number of times it compounds per year.

12. Find the amount that results from the investment: $50 invested at 6% compounded monthly after a period of 3 years. EXAMPLE $59.83

13. Investing $1,000 at a rate of 10% compounded annually, quarterly, monthly, and daily will yield the following amounts after 1 year: FV = PV(1 + r) = 1,000(1 +.1) = $1100.00 COMPARING COMPOUNDING PERIODS

14. Investing $1,000 at a rate of 10% compounded annually, quarterly, monthly, and daily will yield the following amounts after 1 year: COMPARING COMPOUNDING PERIODS

15. Investing $1,000 at a rate of 10% compounded annually, quarterly, monthly, and daily will yield the following amounts after 1 year: Interest Earned

16. Homework

17. Sum of a Finite Geometric Series The sum of the first n terms of a geometric series is Notice – no last term needed!!!!

18. Formula for the Sum of a Finite Geometric Series n = # of terms a 1 = 1 st term r = common ratio What is n? What is a 1 ? What is r?

19. Example Find the sum of the 1 st 10 terms of the geometric sequence: 2,-6, 18, -54 What is n? What is a 1 ? What is r? That’s It!

20. Example: Consider the geometric series 4+2+1+½+…. Find the sum of the first 10 terms.

21. G e o m e t r i c S e r i e s

22. Infinite Geometric Series Consider the infinite geometric sequence What happens to each term in the series? They get smaller and smaller, but how small does a term actually get? Each term approaches 0

23. Infinite Sum 1 1 What is the area of the square? Cut the square in half and label the area of one section. Cut the unlabeled area in half and label the area of one section. Continue the process… Sum all of the areas: The general term is… Since the infinite sum represents the area of the square…

24. Infinite Series

25. Connecting Series and Sequences Find the sum of…

26. Partial Sums of a Series

27. Convergent or Divergent Series

28. Convergent or Divergent Series

29. Examples Why do 1, 3 Diverge?

34. Arithmetic and Geometric Series

35. Definition of a Geometric Series

36. Partial Sums Look at the sequence of partial sums: What is happening to the sum? It is approaching 1 0 1 It’s CONVERGING TO 1.

37. Here’s the Rule Sum of an Infinite Geometric Series If |r| < 1, the infinite geometric series a 1 + a 1 r + a 1 r 2 + … + a 1 r n + … converges to the sum If |r| > 1, then the series diverges (does not have a sum)

38. Converging – Has a Sum So, if -1 < r < 1, then the series will converge. Look at the series given by Since r =, we know that the sum is The graph confirms:

39. Diverging – Has NO Sum If r > 1, the series will diverge. Look at 1 + 2 + 4 + 8 + …. Since r = 2, we know that the series grows without bound and has no sum. The graph confirms:

40. Example Find the sum of the infinite geometric series 9 – 6 + 4 - … We know: a 1 = 9 and r = ?

41. You Try Find the sum of the infinite geometric series 24 – 12 + 6 – 3 + … Since r = -½