Holt Algebra 2 12-4 Geometric Sequences and Series 12-4 Geometric Sequences and Series Holt Algebra 2 Warm Up Warm Up Lesson Presentation Lesson Presentation

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

Holt Algebra 2

12-4 Geometric Sequences and Series

Warm UpSimplify.

1. 2.

3. (–2)8 4.

Solve for x.

5.

Evaluate.

96

256

Holt Algebra 2


Find terms of a geometric sequence, including geometric means.

Find the sums of geometric series.

Objectives

Holt Algebra 2


geometric sequencegeometric meangeometric series

Vocabulary

Holt Algebra 2


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.

Holt Algebra 2


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 .

Holt Algebra 2


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.

Holt Algebra 2


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.

Holt Algebra 2



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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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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Check It Out! Example 1a


Differences

It could be geometric with

Ratios

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1.7, 1.3, 0.9, 0.5, . . .

Check It Out! Example 1b


1.7 1.3 0.9 0.5 Differences –0.4 –0.4 –0.4

It could be arithmetic, with r = –0.4.

Ratio

Holt Algebra 2


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

Holt Algebra 2


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

Holt Algebra 2


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.

Holt Algebra 2


Holt Algebra 2


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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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.

Holt Algebra 2


Check Extend the sequence.

a4 = 192

a5 = 192(4) = 768

a6 = 768(4) = 3072

a7 = 3072(4) = 12,288

Given

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Find the 9th term of the geometric sequence.


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Check It Out! Example 2a Continued


an = a1 r n–1 General rule

The 9th term is .

Substitute for a1, 9 for

n, and for r.

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Given

a6 =

a7 =

a8 =

a9 =

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0.001, 0.01, 0.1, 1, 10, . . .


Find the 9th term of the geometric sequence.


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Check It Out! Example 2b Continued


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.

Holt Algebra 2




a6 = 10(10) = 100

a7 = 100(10) = 1,000

a8 = 1,000(10) = 10,000

a9 = 10,000(10) = 100,000

Givena5 = 10

Holt Algebra 2


Find the 8th term of the geometric sequence with a3 = 36 and a5 = 324.

Example 3: Finding the nth Term Given Two Terms


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.

Holt Algebra 2


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

Holt Algebra 2


Example 3 Continued

Step 3 Write the rule and evaluate for a8.


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

Holt Algebra 2


When given two terms of a sequence, be sure to consider positive and negativevalues for r when necessary.

Caution!

Holt Algebra 2



Find the 7th term of the geometric sequence with the given terms.

a4 = –8 and a5 = –40


a5 = a4 r(5 – 4)

a5 = a4 r

–40 = –8r

5 = r


Simplify.

Substitute –40 for a5 and –8 for a4.

Divide both sides by –8.

Holt Algebra 2



Step 2 Find a1.

an = a1r n - 1

–8 = a1(5)4 - 1

–0.064 = a1

General rule

Use a5 = –8 and r = 5.

Holt Algebra 2




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.

Holt Algebra 2


a2 = 768 and a4 = 48


Find the 7th term of the geometric sequence with the given terms.


a4 = a2 r(4 – 2)

a4 = a2 r2

48 = 768r2

0.0625 = r2


Simplify.

Substitute 48 for a4 and 768 for a2.

Divide both sides by 768.

±0.25 = r Take the square root.

Holt Algebra 2



Step 2 Find a1.


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

Holt Algebra 2





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

Holt Algebra 2



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

Holt Algebra 2


Geometric means are the terms between any two nonconsecutive terms of a geometric sequence.

Holt Algebra 2


Example 4: Finding Geometric Means

Use the formula.

Find the geometric mean of and .

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Check It Out! Example 4

Find the geometric mean of 16 and 25.

Use the formula.

Holt Algebra 2


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.

Holt Algebra 2


Holt Algebra 2


Find the indicated sum for the geometric series.

Example 5A: Finding the Sum of a Geometric Series

S8 for 1 + 2 + 4 + 8 + 16 + ...


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Example 5A Continued

Step 2 Find S8 with a1 = 1, r = 2, and n = 8.

Sum

formula

Substitute.

Check Use a graphing calculator.

Holt Algebra 2


Example 5B: Finding the Sum of a Geometric Series

Find the indicated sum for the geometric series.

Step 1 Find the first term.

Holt Algebra 2


Example 5B Continued

Step 2 Find S6.

= 1(1.96875) ≈ 1.97

Check Use a graphing calculator.

Sum

formula

Substitute.

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Find the indicated sum for each geometric series.


S6 for

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Step 2 Find S6 with a1 = 2, r = , and n = 6.

Substitute.

Sum formula

Holt Algebra 2



Find the indicated sum for each geometric series.

Step 1 Find the first term.

Holt Algebra 2


Step 2 Find S6.


Holt Algebra 2


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.

Holt Algebra 2


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

Holt Algebra 2


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.

Holt Algebra 2


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

Holt Algebra 2


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

Holt Algebra 2


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

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Holt Algebra 2 12-4 Geometric Sequences and Series 12-4 Geometric Sequences and Series Holt Algebra 2 Warm Up Warm Up Lesson Presentation Lesson Presentation