1. SECTION 7.3 GEOMETRIC SEQUENCES

2. (a) 3, 6, 12, 24, 48,96 (b) 12, 4, 4/3, 4/9, 4/27, 4/27,4/81 (c).2,.6, 1.8, 5.4, 16.2, 16.2,48.6 Geometric Sequences have a “common ratio”. (a) r = 2 (b) r = 1/3 (c) r = 3

3. GEOMETRIC SEQUENCE RECURSION FORMULA a n = ra n - 1 This formula relates each term in the sequence to the previous term in the sequence. a n = 2a n - 1 b n = 1/3b n - 1 c n = 3c n - 1

4. EXAMPLE: Given that a 1 = 5 and the recursion formula a n = 1.5a n - 1, determine the the value of a 5. a 2 = 1.5(5) = 7.5 a 3 = 1.5(7.5) = 11.25 a 4 = 1.5(11.25) = 16.875 a 5 = 1.5(16.875) = 25.3125

5. Again, recursion formulas have a big disadvantage! Explicit Formulas are much better for finding nth terms.

6. a 2 = ra 1 a 3 = ra 2 = r(ra 1 ) = r 2 a 1 a 4 = ra 3 = r(r 2 a 1 ) = r 3 a 1 In general, a n = r n - 1 a 1 GEOMETRIC SEQUENCE EXPLICIT FORMULA

7. PREVIOUS EXAMPLE: Given that a 1 = 5 and r = 1.5, determine the the value of a 5. a 5 = 1.5 4 (5) = 25.3125

8. EXAMPLE: Given that {a n } = 64, 48, 36... determine the value of a 8 First, determine r r = 48/64 =.75 a 8 =.75 7 (64) a8 =a8 =a8 =a8 =

9. EXAMPLE: If a person invests $500 today at 6% interest compounded monthly, how much will the investment be worth at the end of 10 years (that is, at the end of 120 months)? The 6% is an annual rate. The corresponding monthly rate is.06/12 =.005

10. EXAMPLE: a 1 = 500(1.005) Amt at end of mth 1 a 2 = 500(1.005) 2 Amt at end of mth 2   a 120 = 500(1.005) 120 Amt at end of mth 120

11. EXAMPLE: a 120 = 500(1.005) 120 Amt at end of mth 120 $909.70

12. GEOMETRIC SEQUENCE SUM FORMULA Let a 1, a 2, a 3 be a geometric sequence Then S n = a 1 + a 2 + a 3 +... + a n is the sum of the first n terms of that sequence. S n can also be written as S n = a 1 + a 1 r + a 1 r 2 +... + a 1 r n - 1

13. GEOMETRIC SEQUENCE SUM FORMULA S n = a 1 + a 1 r + a 1 r 2 +... + a 1 r n - 1 S n = a 1 + a 1 r + a 1 r 2 +... + a 1 r n - 1 rS n = a 1 r + a 1 r 2 +... + a 1 r n - 1 + a 1 r n S n - rS n = a 1 + 0 + 0 +... + 0 + - a 1 r n S n (1 - r) = a 1 (1 - r n )

14. EXAMPLE: Determine the sum of the first 20 terms of the geometric sequence 36, 12, 4, 4/3,... a 1 = 36r = 1/3

15. EXAMPLE: 53.9999999

16. If you were offered 1¢ today, 2¢ tomorrow, 4¢ the third day and so on for 20 days or a lump sum of $10,000, which would you choose? = $10,485.75

17. This formula is for the sum of the first n terms of a geometric sequence. Can we find the sum of an entire sequence? For example:1 + 3 + 9 + 27 +...

18. SUMS OF ENTIRE GEOMETRIC SEQUENCES But we can for a sequence such as 1 as n 

19. GEOMETRIC SEQUENCE SUM FORMULA Any geometric sequence with  r  < 1 As n  ,  r  < 1

20. EXAMPLE: Evaluate the sum of the geometric series: 16 + 12 + 9 + 27/4 +... r = 3/4 64

21. EXAMPLE: A ball is dropped from a height of 16 feet. At each bounce it rises to a height of three-fourths the previous height. How far will it have traveled (up and down) by the time it comes to rest?

22. EXAMPLE: Down series:16 + 12 + 9 +... Up series:12 + 9 + 27/4... 64 + 48 = 112 ft. 64 + 48 = 112 ft.

23. Geometric Series

24. EXAMPLE 3 + 3 2 + 3 3 +... + 3 8 Geometric Sequence

25. CONCLUSION OF SECTION 7.3 CONCLUSION OF SECTION 7.3