12-4 Geometric Sequences and Series. Lesson 12 3 quiz Battle of the CST s Lesson Presentation

Size: px

Start display at page:

Download "12-4 Geometric Sequences and Series. Lesson 12 3 quiz Battle of the CST s Lesson Presentation"

Amanda Moody
5 years ago
Views:

1 12-4 Geometric Sequences and Series Lesson 12 3 quiz Battle of the CST s Lesson Presentation

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

3 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.

4 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.

5 Recall that exponential functions have a common ratio. When you graph the ordered pairs (n, a n ) 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.

6 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, , 93, 86, 79 Differences Ratios It could be arithmetic, with d = 7.

7 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, , 90, 60, 15 Differences Ratios It is neither.

8 Check for understanding 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 Ratios It could be geometric, with

9 Check for Understanding Determine whether the sequence could be geometric or arithmetic. If possible, find the common ratio or difference. Differences Ratios It could be geometric with

10 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. nth term a n = a n 1 r Common ratio First term

12 Example 2A: 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. r = a 2 12 = a 1 3 = 4 Step 2 Write a rule, and evaluate for n = 7. a n = a 1 r n 1 General rule a 7 = 3(4) 7 1 Substitute 3 for a 1, 7 for n, and 4 for r. = 3(4096) = 12,288 The 7th term is 12,288.

13 Check for Understanding Find the 9th term of the geometric sequence. Step 1 Find the common ratio. Step 2 Write a rule, and evaluate for n = 9. a n = a 1 r n The 9th term is. General rule Substitute n, and for r. for a 1, 9 for

14 Example 3A: Finding the nth Term Given Two Terms Find the 8th term of the geometric sequence with a 3 = 36 and a 5 = 324. Step 1 Find the common ratio. a 5 = a 3 r (5 3) Use the given terms. a 5 = a 3 r = 36r 2 Simplify. Substitute 324 for a 5 and 36 for a 3. 9 = r 2 3 = r Divide both sides by 36. Take the square root of both sides. Step 2 Find a 1. Consider both the positive and negative values for r. a n = a 1 r n - 1 a n = a 1 r n - 1 General rule 36 = a 1 (3) 3-1 or 36 = a 1 ( 3) 3-1 Use a 3 = 36 and r = 3. 4 = a 1 4 = a 1

15 Example 3A Continued Step 3 Write the rule and evaluate for a 8. Consider both the positive and negative values for r. a n = a 1 r n - 1 a n = a 1 r n - 1 General rule a n = 4(3) n - 1 or a n = 4( 3) n - 1 Substitute a 1 and r. a 8 = 4(3) 8-1 a 8 = 8748 The 8th term is 8748 or a 8 = 4( 3) 8-1 a 8 = 8748 Evaluate for n = 8. Caution! When given two terms of a sequence, be sure to consider positive and negative values for r when necessary.

16 Check for Understanding Find the 7th term of the geometric sequence with the given terms. a 4 = 8 and a 5 = 40 Step 1 Find the common ratio. a 5 = a 4 r (5 4) a 5 = a 4 r 40 = 8r 5 = r Step 2 Find a 1. a n = a 1 r n - 1 Use the given terms. Simplify. Substitute 40 for a 5 and 8 for a 4. Divide both sides by 8. General rule 8 = a 1 (5) 4-1 Use a 5 = 8 and r = = a 1

17 Check for understanding continued Step 3 Write the rule and evaluate for a 7. a n = a 1 r n - 1 a n = 0.064(5) n - 1 a 7 = 0.064(5) 7-1 a 7 = 1,000 Substitute for a 1 and r. Evaluate for n = 7. The 7th term is 1,000.

18 END OF DAY ONE: Homework: Pg. 895 #2 10 (all) Begin Part 2 Homework Day 2: Pg. 895 #11 17, (odd) Lesson Quiz Tomorrow

19 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 S n and r from S n as shown.

21 Example 5A: Finding the Sum of a Geometric Series Find the indicated sum for the geometric series. S 8 for Step 1 Find the common ratio.

22 Example 5A Continued Step 2 Find S 8 with a 1 = 1, r = 2, and n = 8. Sum formula Substitute.

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

24 Example 5B Continued Step 2 Find S 6. Sum formula Substitute. = 1( ) 1.97

25 Check for understanding Find the indicated sum for each geometric series. S 6 for Step 1 Find the common ratio.

26 Example 5C Continued Step 2 Find S 6 with a 1 = 2, r =, and n = 6. Sum formula Substitute.

27 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, a n = the number of games played in the nth round, and S n = the total number of games played through n rounds.

28 Example 6 Continued Step 2 Find the number of rounds required. The final round will have 1 game, so substitute 1 for a n. Isolate the exponential expression by dividing by = n 1 5 = n Equate the exponents. Solve for n.

29 Example 6 Continued Step 3 Find the total number of games after 5 rounds. Sum function for geometric series 341 games must be played to determine the winner.

30 Check for understanding A 6-year lease states that the annual rent for an office space is $84,000 the first year and will increase by 8% each additional year of the lease. What will the total rent expense be for the 6-year lease? $616, End of Day 2. Lesson Quiz tomorrow.

Geometric Sequences. Geometric Sequences. Warm Up Lesson Presentation Lesson Quiz. Holt McDougal Algebra 1

Geometric Sequences. Geometric Sequences. Warm Up Lesson Presentation Lesson Quiz. Holt McDougal Algebra 1 Warm Up Lesson Presentation Lesson Quiz Algebra 1 Warm Up Find the value of each expression. 1. 2 5 32 2. 2 5 3. 3 4 81 4. ( 3) 4 81 5. (0.2) 3 0.008 6. 7( 4) 2 112 7. 8. 12( 0.4) 3 0.768 Objectives Recognize