holt algebra 2 12-4 geometric sequences and series 12-4 geometric sequences and series holt algebra...
TRANSCRIPT
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Holt Algebra 2
12-4 Geometric Sequences and Series12-4 Geometric Sequences and Series
Holt Algebra 2
Warm UpWarm Up
Lesson PresentationLesson Presentation
Lesson QuizLesson Quiz
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Holt Algebra 2
12-4 Geometric Sequences and Series
Warm UpSimplify.
1. 2.
3. (–2)8 4.
Solve for x.
5.
Evaluate.
96
256
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Holt Algebra 2
12-4 Geometric Sequences and Series
Find terms of a geometric sequence, including geometric means.
Find the sums of geometric series.
Objectives
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Holt Algebra 2
12-4 Geometric Sequences and Series
geometric sequencegeometric meangeometric series
Vocabulary
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Holt Algebra 2
12-4 Geometric Sequences and Series
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.
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Holt Algebra 2
12-4 Geometric Sequences and Series
The number of players remaining after each round can be modeled by a geometric sequence. In a geometric sequence, the ratio of successiveterms is a constant called the common ratio r (r ≠ 1) . For the players remaining, r is .
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Holt Algebra 2
12-4 Geometric Sequences and Series
Recall that exponential functions have a commonratio. When you graph the ordered pairs (n, an) of ageometric sequence, the points lie on an exponentialcurve as shown. Thus, you can think of a geometricsequence as an exponential function with sequentialnatural numbers as the domain.
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Holt Algebra 2
12-4 Geometric Sequences and Series
Determine whether the sequence could be geometric or arithmetic. If possible, find the common ratio or difference.
Example 1A: Identifying Geometric Sequences
100, 93, 86, 79, ...
100, 93, 86, 79
Differences –7 –7 –7
Ratios 93 86 79 100 93 86
It could be arithmetic, with d = –7.
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Holt Algebra 2
12-4 Geometric Sequences and Series
Determine whether the sequence could be geometric or arithmetic. If possible, find the common ratio or difference.
Example 1B: Identifying Geometric Sequences
180, 90, 60, 15, ...
180, 90, 60, 15
Differences –90 –30 –45
It is neither.
3Ratios 1 1 1
2 4
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Holt Algebra 2
12-4 Geometric Sequences and Series
Determine whether the sequence could be geometric or arithmetic. If possible, find the common ratio or difference.
Example 1C: Identifying Geometric Sequences
5, 1, 0.2, 0.04, ...
5, 1, 0.2, 0.04
Differences –4 –0.8 –0.16
5Ratios 1 1 1
5 5
It could be geometric, with
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Holt Algebra 2
12-4 Geometric Sequences and Series
Check It Out! Example 1a
Determine whether the sequence could be geometric or arithmetic. If possible, find the common ratio or difference.
Differences
It could be geometric with
Ratios
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Holt Algebra 2
12-4 Geometric Sequences and Series
1.7, 1.3, 0.9, 0.5, . . .
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 Differences –0.4 –0.4 –0.4
It could be arithmetic, with r = –0.4.
Ratio
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Holt Algebra 2
12-4 Geometric Sequences and Series
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, . . . Differences 18 14 10
It is neither.
Ratios
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Holt Algebra 2
12-4 Geometric Sequences and Series
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.
an = an–1r nth termCommon ratio
First term
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Holt Algebra 2
12-4 Geometric Sequences and Series
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.
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Holt Algebra 2
12-4 Geometric Sequences and Series
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Holt Algebra 2
12-4 Geometric Sequences and Series
Find the 7th term of the geometric sequence 3, 12, 48, 192, ....
Example 2: Finding the nth Term Given a Geometric Sequence
Step 1 Find the common ratio.
r = a2
a1
123
= 4=
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Holt Algebra 2
12-4 Geometric Sequences and Series
Example 2 Continued
Step 2 Write a rule, and evaluate for n = 7.
an = a1 r n–1
a7 = 3(4)7–1
= 3(4096) = 12,288
The 7th term is 12,288.
General rule
Substitute 3 for a1,7 for n, and 4 for r.
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Holt Algebra 2
12-4 Geometric Sequences and Series
Check Extend the sequence.
a4 = 192
a5 = 192(4) = 768
a6 = 768(4) = 3072
a7 = 3072(4) = 12,288
Given
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Holt Algebra 2
12-4 Geometric Sequences and Series
Check It Out! Example 2a
Find the 9th term of the geometric sequence.
Step 1 Find the common ratio.
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Holt Algebra 2
12-4 Geometric Sequences and Series
Check It Out! Example 2a Continued
Step 2 Write a rule, and evaluate for n = 9.
an = a1 r n–1 General rule
The 9th term is .
Substitute for a1, 9 for
n, and for r.
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Holt Algebra 2
12-4 Geometric Sequences and Series
Check It Out! Example 2a Continued
Check Extend the sequence.
Given
a6 =
a7 =
a8 =
a9 =
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Holt Algebra 2
12-4 Geometric Sequences and Series
0.001, 0.01, 0.1, 1, 10, . . .
Check It Out! Example 2b
Find the 9th term of the geometric sequence.
Step 1 Find the common ratio.
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Holt Algebra 2
12-4 Geometric Sequences and Series
Check It Out! Example 2b Continued
Step 2 Write a rule, and evaluate for n = 9.
an = a1 r n–1
a9 = 0.001(10)9–1
= 0.001(100,000,000) = 100,000
The 7th term is 100,000.
General rule
Substitute 0.001 for a1,
9 for n, and 10 for r.
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Holt Algebra 2
12-4 Geometric Sequences and Series
Check It Out! Example 2b Continued
Check Extend the sequence.
a6 = 10(10) = 100
a7 = 100(10) = 1,000
a8 = 1,000(10) = 10,000
a9 = 10,000(10) = 100,000
Givena5 = 10
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Holt Algebra 2
12-4 Geometric Sequences and Series
Find the 8th term of the geometric sequence with a3 = 36 and a5 = 324.
Example 3: Finding the nth Term Given Two Terms
Step 1 Find the common ratio.
a5 = a3 r(5 – 3)
a5 = a3 r2
324 = 36r2
9 = r2
3 = r
Use the given terms.
Simplify.
Substitute 324 for a5 and 36 for a3.
Divide both sides by 36.
Take the square root of both sides.
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Holt Algebra 2
12-4 Geometric Sequences and Series
Example 3 Continued
Step 2 Find a1.
Consider both the positive and negative values for r.
an = a1r n - 1
36 = a1(3)3 - 1
4 = a1
an = a1r n - 1
36 = a1(–3)3 - 1
4 = a1
General rule
Use a3 = 36 and r = 3.
or
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Holt Algebra 2
12-4 Geometric Sequences and Series
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
Substitute a1 and r.
The 8th term is 8748 or –8747.
an = 4(3)n - 1
a8 = 4(3)8 - 1
a8 = 8748
an = 4(–3)n - 1
a8 = 4(–3)8 - 1
a8 = –8748
Evaluate for n = 8.
General rule
or
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Holt Algebra 2
12-4 Geometric Sequences and Series
When given two terms of a sequence, be sure to consider positive and negativevalues for r when necessary.
Caution!
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Holt Algebra 2
12-4 Geometric Sequences and Series
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 = a4 r(5 – 4)
a5 = a4 r
–40 = –8r
5 = r
Use the given terms.
Simplify.
Substitute –40 for a5 and –8 for a4.
Divide both sides by –8.
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Holt Algebra 2
12-4 Geometric Sequences and Series
Check It Out! Example 3a Continued
Step 2 Find a1.
an = a1r n - 1
–8 = a1(5)4 - 1
–0.064 = a1
General rule
Use a5 = –8 and r = 5.
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Holt Algebra 2
12-4 Geometric Sequences and Series
Check It Out! Example 3a Continued
Step 3 Write the rule and evaluate for a7.
an = a1r n - 1
Substitute for a1 and r.
The 7th term is –1,000.
an = –0.064(5)n - 1
a7 = –0.064(5)7 - 1
a7 = –1,000
Evaluate for n = 7.
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Holt Algebra 2
12-4 Geometric Sequences and Series
a2 = 768 and a4 = 48
Check It Out! Example 3b
Find the 7th term of the geometric sequence with the given terms.
Step 1 Find the common ratio.
a4 = a2 r(4 – 2)
a4 = a2 r2
48 = 768r2
0.0625 = r2
Use the given terms.
Simplify.
Substitute 48 for a4 and 768 for a2.
Divide both sides by 768.
±0.25 = r Take the square root.
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Holt Algebra 2
12-4 Geometric Sequences and Series
Check It Out! Example 3b Continued
Step 2 Find a1.
Consider both the positive and negative values for r.
an = a1r n - 1
768 = a1(0.25)2 - 1
3072 = a1
an = a1r n - 1
768 = a1(–0.25)2 - 1
–3072 = a1
General rule
Use a2= 768 and r = 0.25.
or
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Holt Algebra 2
12-4 Geometric Sequences and Series
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
a7 = 3072(0.25)7 - 1
a7 = 0.75
an = 3072(–0.25)n - 1
a7 = 3072(–0.25)7 - 1
a7 = 0.75
Evaluate for n = 7.
or
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Holt Algebra 2
12-4 Geometric Sequences and Series
Check It Out! Example 3b Continued
an = a1r n - 1 an = a1r n - 1
Substitute for a1 and r.
The 7th term is 0.75 or –0.75.
an = –3072(0.25)n - 1
a7 = –3072(0.25)7 - 1
a7 = –0.75
an = –3072(–0.25)n - 1
a7 = –3072(–0.25)7 - 1
a7 = –0.75
Evaluate for n = 7.
or
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Holt Algebra 2
12-4 Geometric Sequences and Series
Geometric means are the terms between any two nonconsecutive terms of a geometric sequence.
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Holt Algebra 2
12-4 Geometric Sequences and Series
Example 4: Finding Geometric Means
Use the formula.
Find the geometric mean of and .
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Holt Algebra 2
12-4 Geometric Sequences and Series
Check It Out! Example 4
Find the geometric mean of 16 and 25.
Use the formula.
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Holt Algebra 2
12-4 Geometric Sequences and Series
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 Sn and r from Sn as shown.
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Holt Algebra 2
12-4 Geometric Sequences and Series
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Holt Algebra 2
12-4 Geometric Sequences and Series
Find the indicated sum for the geometric series.
Example 5A: Finding the Sum of a Geometric Series
S8 for 1 + 2 + 4 + 8 + 16 + ...
Step 1 Find the common ratio.
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Holt Algebra 2
12-4 Geometric Sequences and Series
Example 5A Continued
Step 2 Find S8 with a1 = 1, r = 2, and n = 8.
Sum
formula
Substitute.
Check Use a graphing calculator.
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Holt Algebra 2
12-4 Geometric Sequences and Series
Example 5B: Finding the Sum of a Geometric Series
Find the indicated sum for the geometric series.
Step 1 Find the first term.
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Holt Algebra 2
12-4 Geometric Sequences and Series
Example 5B Continued
Step 2 Find S6.
= 1(1.96875) ≈ 1.97
Check Use a graphing calculator.
Sum
formula
Substitute.
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Holt Algebra 2
12-4 Geometric Sequences and Series
Check It Out! Example 5a
Find the indicated sum for each geometric series.
Step 1 Find the common ratio.
S6 for
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Holt Algebra 2
12-4 Geometric Sequences and Series
Check It Out! Example 5a Continued
Step 2 Find S6 with a1 = 2, r = , and n = 6.
Substitute.
Sum formula
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Holt Algebra 2
12-4 Geometric Sequences and Series
Check It Out! Example 5b
Find the indicated sum for each geometric series.
Step 1 Find the first term.
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Holt Algebra 2
12-4 Geometric Sequences and Series
Step 2 Find S6.
Check It Out! Example 5b Continued
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Holt Algebra 2
12-4 Geometric Sequences and Series
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?
Example 6: Sports Application
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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Holt Algebra 2
12-4 Geometric Sequences and Series
Example 6 Continued
Step 2 Find the number of rounds required.
The final round will have 1 game, so substitute 1 for an.
Isolate the exponential expression by dividing by 256.
Solve for n.
Equate the exponents.
5 = n
4 = n – 1
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Holt Algebra 2
12-4 Geometric Sequences and Series
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.
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Holt Algebra 2
12-4 Geometric Sequences and Series
Check It Out! Example 6
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?
$616,218.04
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Holt Algebra 2
12-4 Geometric Sequences and Series
Lesson Quiz: Part I
1. Determine whether the sequence could be geometric or arithmetic. If possible, find the common ratio or difference.
2. Find the 8th term of the geometric sequence 1, –2, 4, –8, ….
geometric; r = 6
–128
3. Find the 9th term of the geometric sequence with a2 = 0.3 and a6 = 0.00003.
0.00000003
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Holt Algebra 2
12-4 Geometric Sequences and Series
Lesson Quiz: Part II
34. Find the geometric mean of and 18.
5. Find the indicated sum for the geometric series
6. A math tournament begins with 81 students. Students compete in groups of 3, with 1 person from each trio going on to the next round until there is 1 winner. How many matches must be played in order to complete the tournament?
40
40