Holt McDougal Algebra 2 Geometric Sequences and Series Geometric Sequences and Series Holt Algebra 2 Warm Up Warm Up Lesson Presentation Lesson Presentation Lesson Quiz Lesson Quiz Holt McDougal Algebra 2
Dec 27, 2015
Holt McDougal Algebra 2
Geometric Sequences and SeriesGeometric Sequences and Series
Holt Algebra 2
Warm UpWarm Up
Lesson PresentationLesson Presentation
Lesson QuizLesson Quiz
Holt McDougal Algebra 2
Holt McDougal Algebra 2
Geometric Sequences and Series
Warm UpSimplify.
1. 2.
3. (–2)8 4.
Solve for x.
5.
Evaluate.
96
256
Holt McDougal Algebra 2
Geometric Sequences and Series
Find terms of a geometric sequence, including geometric means.
Find the sums of geometric series.
Objectives
Holt McDougal Algebra 2
Geometric Sequences and Series
geometric sequencegeometric meangeometric series
Vocabulary
Holt McDougal Algebra 2
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.
Holt McDougal Algebra 2
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 .
Holt McDougal Algebra 2
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.
Holt McDougal Algebra 2
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.
Holt McDougal Algebra 2
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
Holt McDougal Algebra 2
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
Holt McDougal Algebra 2
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
Holt McDougal Algebra 2
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
Holt McDougal Algebra 2
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.
Holt McDougal Algebra 2
Geometric Sequences and Series
Check It Out! Example 2a
Find the 9th term of the geometric sequence.
Step 1 Find the common ratio.
Holt McDougal Algebra 2
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.
Holt McDougal Algebra 2
Geometric Sequences and Series
Check It Out! Example 2a Continued
Check Extend the sequence.
Given
a6 =
a7 =
a8 =
a9 =
Holt McDougal Algebra 2
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.
Holt McDougal Algebra 2
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
Holt McDougal Algebra 2
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
Holt McDougal Algebra 2
Geometric Sequences and Series
When given two terms of a sequence, be sure to consider positive and negativevalues for r when necessary.
Caution!
Holt McDougal Algebra 2
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.
Holt McDougal Algebra 2
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.
Holt McDougal Algebra 2
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.
Holt McDougal Algebra 2
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.
Holt McDougal Algebra 2
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
Holt McDougal Algebra 2
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
Holt McDougal Algebra 2
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
Holt McDougal Algebra 2
Geometric Sequences and Series
Geometric means are the terms between any two nonconsecutive terms of a geometric sequence.
Holt McDougal Algebra 2
Geometric Sequences and Series
Example 4: Finding Geometric Means
Use the formula.
Find the geometric mean of and .
Holt McDougal Algebra 2
Geometric Sequences and Series
Check It Out! Example 4
Find the geometric mean of 16 and 25.
Use the formula.
Holt McDougal Algebra 2
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.
Holt McDougal Algebra 2
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.
Holt McDougal Algebra 2
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.
Holt McDougal Algebra 2
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.
Holt McDougal Algebra 2
Geometric Sequences and Series
Example 5B Continued
Step 2 Find S6.
= 1(1.96875) ≈ 1.97
Check Use a graphing calculator.
Sum
formula
Substitute.
Holt McDougal Algebra 2
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
Holt McDougal Algebra 2
Geometric Sequences and Series
Check It Out! Example 5a Continued
Step 2 Find S6 with a1 = 2, r = , and n = 6.
Substitute.
Sum formula
Holt McDougal Algebra 2
Geometric Sequences and Series
Check It Out! Example 5b
Find the indicated sum for each geometric series.
Step 1 Find the first term.
Holt McDougal Algebra 2
Geometric Sequences and Series
Step 2 Find S6.
Check It Out! Example 5b Continued
Holt McDougal Algebra 2
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.
Holt McDougal Algebra 2
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
Holt McDougal Algebra 2
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.