Upload
ethan-mckinney
View
301
Download
0
Embed Size (px)
7.3 Analyze Geometric 7.3 Analyze Geometric Sequences & SeriesSequences & Series
What is a geometric sequence?What is the rule for a geometric sequence?How do you find the nth term given 2 terms?
Geometric SequenceGeometric Sequence
• The ratio of any term to it’s previous The ratio of any term to it’s previous term is constant.term is constant.
• This means you multiply by the same This means you multiply by the same number to get each term.number to get each term.
• This number that you multiply by is This number that you multiply by is called the called the common ratiocommon ratio (r). (r).
ExampleExample: Decide whether each : Decide whether each sequence is geometric.sequence is geometric.
• 4,-8,16,-32,…
• -8/4=-2
• 16/-8=-2
• -32/16=-2
• Geometric (common ratio is -2)
• 3, 9, -27,-81, 243,…
• 9/3 = 3
• -27/9= −3
• -81/-27= 3
• 243/-81=−3
• Not geometric
Tell whether the sequence is geometric. Explain why or why not.
1. 81, 27, 9, 3, 1, . . .
SOLUTION
To decide whether a sequence is geometric, find the ratios of consecutive forms.a2
a1=
27 81 = 3
9 = 13
a3
a2=
9 27 = 3
9 = 13
a4
a3= 3
9 = 13
a5
a4= 1
3
Each ratio is , So the sequence isgeometric.
13
ANSWER
2. 1, 2, 6, 24, 120, . . .
SOLUTION
To decide whether a sequence is geometric find the ratios of consecutive terms.
a2
a1= 2
1 = 2
a3
a2= 6
2 = 3
a4
a3= 24
6 = 4
a5
a4= 120
24 = 5
ANSWER The ratios are different. The sequence isnot geometric.
Rule for a Geometric SequenceRule for a Geometric Sequence
ExampleExample: Write a rule for the nth term of the : Write a rule for the nth term of the sequence 5, 2, 0.8, 0.32,… . Then find asequence 5, 2, 0.8, 0.32,… . Then find a88..
•First, find r.First, find r.
•r= r= 22//5 5 = .4= .4
•aann=5(.4)=5(.4)n-1n-1
aa88=5(.4)=5(.4)8-18-1
aa88=5(.4)=5(.4)77
aa88=5(.0016384)=5(.0016384)
aa88=.008192=.008192
Write a rule for the nth term of the sequence. Then find a7.
a. 4, 20, 100, 500, . . .
SOLUTION
The sequence is geometric with first term a1 = 4 and common ratio
a.
r = 204 = 5. So, a rule for the nth term is:
an = a1 r n – 1
= 4(5)n – 1
Write general rule.
Substitute 4 for a1 and 5 for r.
The 7th term is a7 = 4(5)7 – 1 = 62,500.
Write a rule for the nth term of the sequence. Then find a7.
b. 152, – 76, 38, – 19, . . .
SOLUTION
The sequence is geometric with first term a1 = 152 and common ratio
b.
r = –76152 = – 1
2.So, a rule for the nth term is:
an = a1 r n – 1 Write general rule.
Substitute 152 for a1 and for r.
12
–
198=
7.3 Assignment, Day 17.3 Assignment, Day 1
Page 454, 4-26 evenPage 454, 4-26 even
Geometric Sequences and Seriesday 2
How do you find the nth term given 2 terms?
What is the formula for finding the sum of an finite geometric series?
One term of a geometric sequence is a4 =12. The common ratio is r = 2.
a. Write a rule for the nth term.
SOLUTION
a. Use the general rule to find the first term.
an = a1r n – 1
a4 = a1r 4 – 1
12 = a1(2)3
1.5 = a1
Write general rule.
Substitute 4 for n.
Substitute 12 for a4 and 2 for r.Solve for a1.
Write a rule given a term and the common ratio
an = a1r n – 1
So, a rule for the nth term is:
= 1.5(2) n – 1
Write general rule.
Substitute 1.5 for a1 and 2 for r.
One term of a geometric sequence is a4 =12. The common ratio is r = 2.
b. Graph the sequence.
SOLUTION
Write a rule given a term and the common ratio
Create a table of values for the sequence. The graph of the first 6 terms of the sequence is shown. Notice that the points lie on an exponential curve. This is true for any geometric sequence with r > 0.
b.
ExampleExample: One term of a geometric sequence : One term of a geometric sequence is ais a44=3. The common ratio is r=3. Write a rule =3. The common ratio is r=3. Write a rule
for the nth term. Then graph the sequence.for the nth term. Then graph the sequence.
• If aIf a44=3, then when n=4, =3, then when n=4,
aann=3.=3.
• Use aUse ann=a=a11rrn-1n-1
3=a3=a11(3)(3)4-14-1
3=a3=a11(3)(3)33
3=a3=a11(27)(27)11//99=a=a11
• aann=a=a11rrn-1n-1
aann=(=(11//99)(3))(3)n-1n-1
• To graph, graph the To graph, graph the points of the form points of the form (n,a(n,ann).).
• Such as, (1,Such as, (1,11//99), ),
(2,(2,11//33), (3,1), (4,3),…), (3,1), (4,3),…
Two terms of a geometric sequence are a3 = 248 and a6 = 3072. Find a rule for the nth term.SOLUTION
a3 = a1r 3 – 1
a6 = a1r 6 – 1
– 48 = a1 r 2
3072 = a1r 5
Equation 1
Equation 2
Write a system of equations using an 5 a1r n – 1 and substituting 3 for n (Equation 1) and then 6 for n (Equation 2).
STEP 1
STEP 2 Solve the system.– 48
r 2 = a1
3072 =– 48r 2 (r 5 )
3072 = – 48r 3
–4 = r
– 48 = a1(– 4)2
– 3 = a1
Solve Equation 1 for a1.
Substitute for a1 in Equation 2.
Simplify.
Solve for r.
Substitute for r in Equation 1.
Solve for a1.
STEP 3an = a1r n – 1
an = – 3(– 4)n – 1
Write general rule.
Substitute for a1 and r.
–4 = r – 3 = a1
Example: Two terms of a geometric sequence are Example: Two terms of a geometric sequence are aa22=-4 and a=-4 and a66=-1024. Write a rule for the nth term.=-1024. Write a rule for the nth term.
• Write 2 equations, one for each given term.
a2=a1r2-1 OR -4=a1r
a6=a1r6-1 OR -1024=a1r5
• Use these 2 equations & substitution to solve for a1 & r.
-4/r=a1
-1024=(-4/r)r5
-1024=-4r4
256=r4
4=r & -4=r
If r=4, then a1=-1.
an=(-1)(4)n-1
If r=-4, then a1=1.
an=(1)(-4)n-1
an=(-4)n-1
Both Both Work!Work!
Formula for the Sum of a Finite Formula for the Sum of a Finite Geometric SeriesGeometric Series
r
raS
n
n 1
11
n = # of termsn = # of terms
aa1 1 = 1= 1stst term term
r = common ratior = common ratio
Example: Consider the geometric Example: Consider the geometric series 4+2+1+½+… .series 4+2+1+½+… .
• Find the sum of the first 10 terms.
• Find n such that Sn=31/4.
r
raS
n
n 1
11
21
1
21
14
10
10S
128
1023
1024
20464
21
10241023
4
2110241
1410
S
21
1
21
14
4
31
n
21
1
21
14
4
31
n
2121
14
4
31
n
n
2
118
4
31
n
2
11
32
31n
2
1
32
1n
2
1
32
1
5n
n
n
2
1
32
1
n232 log232=n
Find the sum of the geometric series16
i = 14(3)i – 1.
a1 = 4(3)1– 1 = 4
r = 3
= 86,093,440
Identify first term.
Identify common ratio.
Write rule for S16.
Substitute 4 for a1 and 3 for r.
Simplify.
ANSWER
The sum of the series is 86,093,440.
Movie Revenue
In 1990, the total box office revenue at U.S. movie theaters was about $5.02 billion. From 1990 through 2003, the total box office revenue increased by about 5.9% per year.
Write a rule for the total box office revenue an (in billions of dollars) in terms of the year. Let n = 1 represent 1990.
a.
SOLUTION
Because the total box office revenue increased by the same percent each year, the total revenues from year to year form a geometric sequence. Use a1 = 5.02 and r = 1 + 0.059 = 1.059 to write a rule for the sequence.
a.
an = 5.02(1.059)n – 1 Write a rule for an.
Movie Revenue
In 1990, the total box office revenue at U.S. movie theaters was about $5.02 billion. From 1990 through 2003, the total box office revenue increased by about 5.9% per year.
What was the total box office revenue at U.S. movie theaters for the entire period 1990–2003?
b.
There are 14 years in the period 1990–2003, so find S14.b.
ANSWER
The total movie box office revenue for the period 1990–2003 was about $105 billion.
7. Find the sum of the geometric series 6( – 2)i–1.
( )1 – r 8
1 – r= 6
a1 = 6( – 2) = 6
SOLUTION
r = – 2Identify first term.
Identify common ratio.
Write rule for S8.
1 – (– 2)8
1 – (– 2)= 6
= 6 – 2553
= – 510ANSWER
1 + 21 – 16= 6
What is the formula for finding the sum of an finite geometric series?
r
raS
n
n 1
11
How do you find the nth term given 2 terms?Write two equations with two unknowns and solve by substitution.
7.3 Assignment, Day 27.3 Assignment, Day 2
p.454p.45428-52 even28-52 even