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5-Minute Check on Chapter 2

Transparency 3-1. 5-Minute Check on Chapter 2. Evaluate 42 - |x - 7| if x = -3 Find 4.1  (-0.5) Simplify each expression 3. 8(-2c + 5) + 9c 4. (36d – 18) / (-9)

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5-Minute Check on Chapter 2

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  1. Transparency 3-1 5-Minute Check on Chapter 2 • Evaluate 42 - |x - 7| if x = -3 • Find 4.1  (-0.5) • Simplify each expression • 3. 8(-2c + 5) + 9c 4. (36d – 18) / (-9) • 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 notgreen? • Which of the following is a true statement Standardized Test Practice: 8/4 < 4/8 -4/8 < -8/4 -4/8 > -8/4 -4/8 > 4/8 A B C D Click the mouse button or press the Space Bar to display the answers.

  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 A. Determine whether the sequence is geometric. 1, 4, 16, 64, 256, … Determine the pattern. 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. Divide the second term by the first. Example 2a Find the next three terms in the geometric sequence. 20, –28, 39.2, … The common factor is –1.4. Use this information to find the next three terms. 20, –28, 39.2 –54.88 76.832 –107.5648 Answer: The next 3 terms are –54.88, 76.832, and –107.5648.

  11. Divide the second term by the first. Example 2b 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 27 20.25 15.1875 Answer: The next three terms are 27, 20.25, and 15.1875.

  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.

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

  14. Simplify. In the sequence, and To find you must first find r. Take the square root of each side. Formula: nth term of a geometric sequence Divide each side by 7. Example 5 Find the geometric mean in the sequence 7, ___, 112. 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 a1 and a common ratio r is given by an = a1r n-1 • Homework: • none

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