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12-4

Geometric Sequences and Series. 12-4. Warm Up. Lesson Presentation. Lesson Quiz. Holt McDougal Algebra 2. Holt Algebra 2. Warm Up Simplify. 1. 2. 3. (–2) 8 4. Solve for x . 5. 96. Evaluate. 256. Objectives. Find terms of a geometric sequence, including geometric means.

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12-4

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  1. Geometric Sequences and Series 12-4 Warm Up Lesson Presentation Lesson Quiz Holt McDougal Algebra 2 Holt Algebra 2

  2. Warm Up Simplify. 1. 2. 3. (–2)8 4. Solve for x. 5. 96 Evaluate. 256

  3. Objectives Find terms of a geometric sequence, including geometric means. Find the sums of geometric series.

  4. Vocabulary geometric sequence geometric mean geometric series

  5. Serena Williams was the winner out of 128 players who began the 2003 Wimbledon Ladies’ Singles Championship. After each match, the winner continues to the next round and the loser is eliminated from the tournament. This means that after each round only half of the players remain.

  6. The number of players remaining after each round can be modeled by a geometric sequence. In a geometric sequence, the ratio of successive terms is a constant called the common ratio r (r ≠ 1) . For the players remaining, r is .

  7. Recall that exponential functions have a common ratio. When you graph the ordered pairs (n, an) of a geometric sequence, the points lie on an exponential curve as shown. Thus, you can think of a geometric sequence as an exponential function with sequential natural numbers as the domain.

  8. Ratios 93 86 79 100 93 86 Example 1A: Identifying Geometric Sequences Determine whether the sequence could be geometric or arithmetic. If possible, find the common ratio or difference. 100, 93, 86, 79, ... 100, 93, 86, 79 Differences –7 –7 –7 It could be arithmetic, with d = –7.

  9. Ratios 1 1 1 2 3 4 Example 1B: Identifying Geometric Sequences Determine whether the sequence could be geometric or arithmetic. If possible, find the common ratio or difference. 180, 90, 60, 15, ... 180, 90, 60, 15 Differences –90 –30 –45 It is neither.

  10. Ratios 1 1 1 5 5 5 It could be geometric, with Example 1C: Identifying Geometric Sequences Determine whether the sequence could be geometric or arithmetic. If possible, find the common ratio or difference. 5, 1, 0.2, 0.04, ... 5, 1, 0.2, 0.04 Differences –4 –0.8 –0.16

  11. Differences Ratios It could be geometric with Check It Out! Example 1a Determine whether the sequence could be geometric or arithmetic. If possible, find the common ratio or difference.

  12. –0.4 –0.4 –0.4 Differences Ratio Check It Out! Example 1b Determine whether the sequence could be geometric or arithmetic. If possible, find the common ratio or difference. 1.7, 1.3, 0.9, 0.5, . . . 1.7 1.3 0.9 0.5 It could be arithmetic, with r = –0.4.

  13. Differences 18 14 10 Ratios Check It Out! Example 1c Determine whether each sequence could be geometric or arithmetic. If possible, find the common ratio or difference. –50, –32, –18, –8, . . . –50, –32, –18, –8, . . . It is neither.

  14. an = an–1r Common ratio nth term First term Each term in a geometric sequence is the product of the previous term and the common ratio, giving the recursive rule for a geometric sequence.

  15. You can also use an explicit rule to find the nth term of a geometric sequence. Each term is the product of the first term and a power of the common ratio as shown in the table. This pattern can be generalized into a rule for all geometric sequences.

  16. a2 12 r = = = 4 a1 3 Example 2: Finding the nth Term Given a Geometric Sequence Find the 7th term of the geometric sequence 3, 12, 48, 192, .... Step 1 Find the common ratio.

  17. Example 2 Continued Step 2 Write a rule, and evaluate for n = 7. an =a1rn–1 General rule Substitute 3 for a1,7 for n, and 4 for r. a7 = 3(4)7–1 = 3(4096) = 12,288 The 7th term is 12,288.

  18. Check Extend the sequence. Given a4 = 192 a5 = 192(4) = 768 a6 = 768(4) = 3072  a7 = 3072(4) = 12,288

  19. Check It Out! Example 2a Find the 9th term of the geometric sequence. Step 1 Find the common ratio.

  20. Substitute for a1, 9 for n, and for r. The 9th term is . Check It Out! Example 2a Continued Step 2 Write a rule, and evaluate for n = 9. an =a1rn–1 General rule

  21. a6 = a7 = a8 = a9 = Check It Out! Example 2a Continued Check Extend the sequence. Given

  22. Check It Out! Example 2b Find the 9th term of the geometric sequence. 0.001, 0.01, 0.1, 1, 10, . . . Step 1 Find the common ratio.

  23. Check It Out! Example 2b Continued Step 2 Write a rule, and evaluate for n = 9. an =a1rn–1 General rule Substitute 0.001 for a1, 9 for n, and 10 for r. a9 = 0.001(10)9–1 = 0.001(100,000,000) = 100,000 The 7th term is 100,000.

  24. Check It Out! Example 2b Continued Check Extend the sequence. a5 = 10 Given a6 = 10(10) = 100 a7 = 100(10) = 1,000 a8 = 1,000(10) = 10,000 a9 = 10,000(10) = 100,000

  25. Example 3: Finding the nth Term Given Two Terms Find the 8th term of the geometric sequence with a3 = 36 and a5 = 324. Step 1 Find the common ratio. a5 = a3r(5 – 3) Use the given terms. a5 = a3r2 Simplify. Substitute 324 for a5 and 36 for a3. 324 = 36r2 9 = r2 Divide both sides by 36. 3 = r Take the square root of both sides.

  26. Example 3 Continued Step 2 Find a1. Consider both the positive and negative values for r. an =a1r n - 1 an =a1r n - 1 General rule 36 = a1(3)3 - 1 or 36 = a1(–3)3 - 1 Use a3 =36 and r = 3. 4 = a1 4 = a1

  27. Example 3 Continued Step 3 Write the rule and evaluate for a8. Consider both the positive and negative values for r. an =a1r n - 1 an =a1r n - 1 General rule an =4(3)n - 1 or an =4(–3)n - 1 Substitute a1 and r. a8= 4(3)8- 1 a8 = 4(–3)8 - 1 Evaluate for n = 8. a8 = 8748 a8 = –8748 The 8th term is 8748 or –8747.

  28. Caution! When given two terms of a sequence, be sure to consider positive and negative values for r when necessary.

  29. Check It Out! Example 3a Find the 7th term of the geometric sequence with the given terms. a4 = –8 and a5 = –40 Step 1 Find the common ratio. a5 = a4r(5 – 4) Use the given terms. a5 = a4r Simplify. –40 = –8r Substitute –40 for a5 and –8 for a4. 5 = r Divide both sides by –8.

  30. Check It Out! Example 3a Continued Step 2 Find a1. an =a1r n - 1 General rule –8 = a1(5)4 - 1 Use a5 = –8 and r = 5. –0.064 = a1

  31. Check It Out! Example 3a Continued Step 3 Write the rule and evaluate for a7. an =a1r n - 1 an =–0.064(5)n - 1 Substitute for a1 and r. a7= –0.064(5)7- 1 Evaluate for n = 7. a7 = –1,000 The 7th term is –1,000.

  32. Check It Out! Example 3b Find the 7th term of the geometric sequence with the given terms. a2 = 768 and a4 = 48 Step 1 Find the common ratio. a4 = a2r(4 – 2) Use the given terms. Simplify. a4 = a2r2 48 = 768r2 Substitute 48 for a4 and 768 for a2. 0.0625 = r2 Divide both sides by 768. ±0.25 = r Take the square root.

  33. Check It Out! Example 3b Continued Step 2 Find a1. Consider both the positive and negative values for r. General rule an =a1r n - 1 an =a1r n - 1 Use a2=768 and r = 0.25. 768 = a1(0.25)2 - 1 or 768 = a1(–0.25)2 - 1 3072 = a1 –3072 = a1

  34. Check It Out! Example 3b Continued Step 3 Write the rule and evaluate for a7. Consider both the positive and negative values for r. an =a1r n - 1 an =a1r n - 1 Substitute for a1 and r. an =3072(0.25)n - 1 or an =3072(–0.25)n - 1 a7= 3072(0.25)7- 1 a7 = 3072(–0.25)7 - 1 Evaluate for n = 7. a7 = 0.75 a7 = 0.75

  35. Check It Out! Example 3b Continued an =a1r n - 1 an =a1r n - 1 Substitute for a1 and r. an =–3072(0.25)n - 1 or an =–3072(–0.25)n - 1 a7= –3072(0.25)7- 1 a7 = –3072(–0.25)7 - 1 Evaluate for n = 7. a7 = –0.75 a7 = –0.75 The 7th term is 0.75 or –0.75.

  36. Geometric meansare the terms between any two nonconsecutive terms of a geometric sequence.

  37. Find the geometric mean of and . Example 4: Finding Geometric Means Use the formula.

  38. Check It Out! Example 4 Find the geometric mean of 16 and 25. Use the formula.

  39. The indicated sum of the terms of a geometric sequence is called a geometric series. You can derive a formula for the partial sum of a geometric series by subtracting the product of Snand r from Snas shown.

  40. S8 for 1 + 2 + 4 + 8 + 16 + ... Example 5A: Finding the Sum of a Geometric Series Find the indicated sum for the geometric series. Step 1 Find the common ratio.

  41. Sum formula Example 5A Continued Step 2 Find S8 with a1 = 1, r = 2, and n = 8. Check Use a graphing calculator. Substitute.

  42. Example 5B: Finding the Sum of a Geometric Series Find the indicated sum for the geometric series. Step 1 Find the first term.

  43. Sum formula Example 5B Continued Step 2 Find S6. Check Use a graphing calculator. Substitute. = 1(1.96875) ≈ 1.97

  44. S6 for Check It Out! Example 5a Find the indicated sum for each geometric series. Step 1 Find the common ratio.

  45. Check It Out! Example 5a Continued Step 2 Find S6 with a1 = 2, r = , and n = 6. Sum formula Substitute.

  46. Check It Out! Example 5b Find the indicated sum for each geometric series. Step 1 Find the first term.

  47. Check It Out! Example 5b Continued Step 2 Find S6.

  48. Example 6: Sports Application An online video game tournament begins with 1024 players. Four players play in each game, and in each game, only the winner advances to the next round. How many games must be played to determine the winner? Step 1 Write a sequence. Let n = the number of rounds, an = the number of games played in the nth round, and Sn = the total number of games played through n rounds.

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