Section 10.3: Geometric Sequences & Series

1. Section 10.3: Geometric Sequences & Series Objectives: -Find nth terms and geometric means of geometric sequences - Find sum of n terms of geometric series and the sums of infinite geometric series

2. Definitions Geometric Sequence: when the ratio between consecutive terms is a constant. Common Ratio: denoted r, the constant ratio between terms. *To find r, divide any term by its previous term. *To find the next term in the sequence, MULTIPLY the given term by the common ratio.

3. Example 1 Determine the common ratio and find the next three terms of the geometric sequence: A) –6, 9, –13.5, … . B) 243n – 729, –81n + 243, 27n – 81, … .

4. Example 1 Determine the common ratio and find the next three terms of the geometric sequence: C) 24, 84, 294, … . D) 8, -2, , … .

5. Explicit Formula A formula for the nth term of a geometric sequence can be used to find any term in the sequence.

6. Example 2: Write an explicit formula and a recursive formula for finding the nth term in the sequence: A) –1, 2, –4, … . B) 3, 16.5, 90.75, …

7. Example 3: A) Find the 11th term of the geometric sequence –122, 115.9, –110.105, … . B) Find the 25th term of the geometric sequence 324, 291.6, 262.44, ….

8. Example 3: C) Find r when a2 = -18 and a5 = D) Find r when a2 = -4 and a6 =-1024

9. Exit Slip • Find the 14th term of the geometric sequence3, –9, 27, … .

10. Example 4: A) REAL ESTATE A couple purchased a home for $225,000. At the end of each year, the value of the home appreciates 3%. Write an explicit formula for finding the value of the home after n years. What is the value of the home after the tenth year?

11. Example 4: B) BOAT Jeremy purchased a boat for $12,500. By the end of each year, the value of the boat depreciates 4%. Write an explicit formula for finding the value of the boat after n years. Then find the value of the boat in 20 years.

12. The graphs of terms of a geometric sequence lie on a curve, as shown. A geometric sequence can be modeled by an exponential function in which the domain is restricted to natural numbers.

13. Geometric Mean • If two nonconsecutive terms of a geometric sequence are known, the terms can be calculated. • General steps to do so: • Find the common ratio using the nth term formula for geometric sequences. • Use r to find the geometric means (next terms)

14. Example 5: A) Write a sequence that has three geometric means between 264 and 1.03125.

15. Example 5: B) Write a geometric sequence that has two geometric means between 20 and 8.4375.

16. Definitions Geometric Series: the sum of the terms of a geometric sequence.

17. Example 6: A) Find the sum of the first eleven terms of the geometric series 4, –6, 9, … . B) Find the sum of the first n terms of a geometric series with a1 = –4, an= –65,536, and r = 2.

18. Example 6: C) Find the sum of the first 8 terms of the geometric series 8 + 36 + 162 + … .

19. Example 7: A) Find B) Find

20. Infinite Geometric Series Recall, that an infinite arithmetic series exists when the sequence converges. An infinite geometric series will exist if .

21. Example 8: If possible, find the sum of the infinite geometric series. A) 24 + 18 + 13.5 + … . B) 0.33 + 0.66 + 1.32 + … . C) D)

22. Example 9: Express the series using sigma notation. Then find the indicated sum. a) 3 + 12 + 48 + … + 3072 b) 0.2 – 1 + 5 - … - 625

23. Example 9: Express the series using sigma notation. Then find the indicated sum. c)

24. Example 10: a) In a geometric sequence a2 = –8 and a7= 8192. Find S10.

25. Example 10: b) In a geometric sequence a3 = 48 and a8= 1536. Find S8.

26. Exit Slip: 1. -6, -1, 4, … Find the explicit formula. 2. 80 + 32 + + … Find the sum of first 7 terms.