1. 5-Minute Check on Chapter 2 Transparency 3-1 Click the mouse button or press the Space Bar to display the answers. 1.Evaluate 42 - |x - 7| if x = -3 2.Find 4.1  (-0.5) Simplify each expression 3. 8(-2c + 5) + 9c 4. (36d – 18) / (-9) 5.A bag of lollipops has 10 red, 15 green, and 15 yellow lollipops. If one is chosen at random, what is the probability that it is not green? 6. Which of the following is a true statement Standardized Test Practice: ACBD 8/4 < 4/8-4/8 < -8/4-4/8 > -8/4-4/8 > 4/8

2. Lesson 10-7 Geometric Sequences

3. Transparency 7 Click the mouse button or press the Space Bar to display the answers.

4. Transparency 7a

5. Objectives Recognize and extend geometric sequences Find geometric means

6. Vocabulary Geometric sequence – Common ratio – Geometric means –

7. Four Step Problem Solving Plan Step 1: Explore the Problem –Identify what information is given (the facts) –Identify what you are asked to find (the question) Step 2: Plan the Solution –Find an equation the represents the problem –Let a variable represent what you are looking for Step 3: Solve the Problem –Plug into your equation and solve for the variable Step 4: Examine the Solution –Does your answer make sense? –Does it fit the facts in the problem?

8. Example 1a Determine the pattern. 1 4 16 64 256 A. Determine whether the sequence is geometric. 1, 4, 16, 64, 256, … In this sequence, each term in found by multiplying the previous term by 4. Answer: This sequence is geometric.

9. Example 1b B. Determine whether the sequence is geometric. 1, 3, 5, 7, 9, 11, … Determine the pattern. 1 3 5 7 9 11 In this sequence, each term is found by adding 2 to the previous term. Answer: This sequence is arithmetic, not geometric.

10. Example 2a The common factor is –1.4. Use this information to find the next three terms. 20, –28, 39.2 Find the next three terms in the geometric sequence. 20, –28, 39.2, … Divide the second term by the first. –54.88 76.832–107.5648 Answer:The next 3 terms are –54.88, 76.832, and –107.5648.

11. Example 2b Divide the second term by the first. 27 20.2515.1875 Answer:The next three terms are 27, 20.25, and 15.1875. Find the next three terms in the geometric sequence. 64, 48, 36, … The common factor is 0.8. Use this information to find the next three terms. 64, 48, 36

12. Example 3 Geography The population of the African country of Liberia was about 2,900,000 in 1999. If the population grows at a rate of about 5% per year, what will the population be in the years 2003, 2004, and 2005? The population is a geometric sequence. The first term is 2,900,000 and the common ratio is 1.05. Answer:The population of Liberia in the years 2003, 2004, and 2005 will be about 3,524,968, 3,701,217, and 3,886,277, respectively. YearPopulation 1999 2,900,000 2000 2,900,000(1.05) or 3,045,000 2001 3,045,000(1.05) or 3,197,250 2002 3,197,250(1.05) or 3,357,112.5 2003 3,357,112.5(1.05) or 3,524,968.1 2004 3,524,968.1(1.05) or 3,701,216.5 2005 3,701,216.5(1.05) or 3,886,277.3

13. Example 4 Find the eighth term of a geometric sequence in which Formula for the nth term of a geometric sequence Answer:The eighth term in the sequence is 15,309.

14. Example 5 Find the geometric mean in the sequence 7, ___, 112. In the sequence, and To find you must first find r. Formula: nth term of a geometric sequence Divide each side by 7. Simplify. Take the square root of each side. If r = 4, the geometric mean is 7(4) or 28. If r = -4, the geometric mean is 7(–4) or –28. Answer:The geometric mean is 28 or –28.

15. Example 5, alternate way Find the geometric mean in the sequence 7, ___, 112. The geometric mean (GM) of two numbers, a and b, is the square root of their product: GM =  a  b. The geometric mean in the sequence, 7, ___, 112 would be (the number between them) GM =  7  112 =  784 =  28. Check: (using r from last page) 7  4 = 28 and 28  4 = 112

16. Summary & Homework Summary: –A geometric sequence is a sequence in which each term after the nonzero first term is found by multiplying the previous term by a constant called the common ratio r, where r ≠ 0 or 1 –The nth term an of a geometric sequence with the first term a 1 and a common ratio r is given by a n = a 1 r n-1 Homework: –none