2. Geometric Sequences & Series 8.3 JMerrill, 2007 Revised 2008

3. Sequences A Sequence: Usually defined to be a function Domain is the set of positive integers Arithmetic sequence graphs are linear (usually) Geometric sequence graphs are exponential

4. Geometric Sequences GEOMETRIC - the ratio of any two consecutive terms in constant. Always take a number and divide by the preceding number to get the ratio 1,3,9,27,81………. ratio = 3 64,-32,16,-8,4…… ratio = -1/2 a,ar,ar2,ar3……… ratio = r

5. What is the ratio of 4, 8, 16, 32… 2

6. What is the ratio of 27, -18, 12,-8… -2/3

7. Is the Sequence 3, 8, 13, 18… A.Arithmetic B.Geometric C.Neither

8. Is the Sequence 2, 5, 10, 17… A.Arithmetic B.Geometric C.Neither

9. Is the Sequence 8, 12, 18, 27… A.Arithmetic B.Geometric C.Neither

10. Example Write the first six terms of the geometric sequence with first term 6 and common ratio 1/3.

11. Formulas for the n th term of a Sequence Geometric:an = = = =a1 * r (n-1) To get the nth term, start with the 1st term and multiply by the ratio raised to the (n-1) power n = THE TERM NUMBER

12. Example Find a formula for an and sketch the graph for the sequence 8, 4, 2, 1... Arithmetic or Geometric? r = ? an = = = = a1 (r (n-1) ) an = = = = 8 * ½ (n-1)

13. Using the Formula Find the 8th term of the geometric sequence whose first term is -4 and whose common ratio is -2 an = = = =a1 * r (n-1) a8 = = = =-4 * (-2) (8-1) a8 = -4(-128) = 512

14. Example Find the given term of the geometric sequence if a3 = 12, a6 =96, find a11 r = ? Since a1 is unknown. Use given info an = = = = a1 * r (n-1)an = = = = a1 * r (n-1) a3 = = = = a1 * r2a6 = = = = a1 * r5 12 = = = = a1 *r296 = a1 *r5

15. Example

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

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

18. 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

19. Partial Sums Look at the sequence of partial sums: What is happening to the sum? It is approaching 1 0 1

20. 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)

21. 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:

22. 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:

23. Example Find the sum of the infinite geometric series 9 – 6 + 4 - … We know: a1 = 9 and r = ?

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

25. Example Ex: The infinite, repeating decimal 0.454545… can be written as the infinite series 0.45 + 0.0045 + 0.000045 + … What is the sum of the series? (Express the decimal as a fraction in lowest terms)

26. You Try Express the repeating decimal, 0.777…, as a rational number (hint: the sum!)

27. You Try, Part Deux Find the first three terms of an infinite geometric sequence with sum 16 and common ratio

28. Last Example Find the following sum: What’s the first term? What’s the second term? Arithmetic or Geometric? What’s the common ratio? Plug into the formula… 12 24 2

29. Can You Do It??? Find the sum, if possible, of 8