Algebra 2 CH 7.3 Analyze Geometric Sequences and Series.notebook
Algebra 2 CH 7.3 Analyze Geometric Sequences and Series.notebook
February 11, 2013
Analyze Geometric Sequences and Series
Analyze Geometric Sequences and Series
In an geometric sequence, the ratio between consecutive terms is constant.
7.3 Analyze Geometric Sequences and Series • Geometric sequence
3, 12, 48, 192, 768, 3072... x4 x4 x4 x4 x4
• Common ratio • Geometric sequence
The ratio is called the common ratio.
x
x
x
x
x
625, 125, 25, 5, 1, 1/5... Jan 2412:50 PM
Jan 2412:50 PM
,
Analyze Geometric Sequences and Series 3, 12, 48, 192, 768, 3072... The common ratio is calculated by dividing a number by the previous number.
Analyze Geometric Sequences and Series Rule for an geometric sequence: an = a1 (r n1 )
What formula that we used in the past does this look like? Jan 2412:50 PM
Analyze Geometric Sequences and Series
an = a1 r n1 Write a rule for the nth term of the sequence:
Jan 2412:50 PM
Analyze Geometric Sequences and Series
1/9, 1/3, 1, 3, 9, 27,... We need to find a1 and r.
a1 = 1/9
r = 3
1/9, 1/3, 1, 3, 9, 27,...
an =1/9 ( 3 n1) Now find the 20 th term.
Now find the 20 th term.
Jan 2412:50 PM
Jan 2412:50 PM
1
Algebra 2 CH 7.3 Analyze Geometric Sequences and Series.notebook
Analyze Geometric Sequences and Series One term of a geometric sequence is a5 = 324 and the common ratio is 6.
February 11, 2013
Analyze Geometric Sequences and Series One term of a geometric sequence is a5 = 324 and the common ratio is 6. Find the a1:
Find the rule for the nth term
an = a1 r n1
Jan 2412:50 PM
Analyze Geometric Sequences and Series
an = a1 r n1
324 = a 1(6 51 )
Jan 2412:50 PM
Analyze Geometric Sequences and Series
324 = a 1(6 51 )
One term of a geometric sequence is a5 = 324 and the common ratio is 6.
324 = a 1(6 4)
Find the rule for the nth term
324 = a 1(1296)
0.25 = a 1
a1=4
r = 6
an = a1 r n1
an =0.25 ( 6 n1) Jan 2412:50 PM
Analyze Geometric Sequences and Series
Jan 2412:50 PM
Analyze Geometric Sequences and Series
One term of an geometric sequence is a3 = 48 and another term is a6=3072.
Find r when a3 = 48 and a6=3072.
Find the rule for the nth term
I. Divide the latter term by the earlier term:
an = a1 r n1
Jan 2412:50 PM
Jan 2412:50 PM
2
Algebra 2 CH 7.3 Analyze Geometric Sequences and Series.notebook
Analyze Geometric Sequences and Series Find r when a3 = 48 and a6=3072.
February 11, 2013
Analyze Geometric Sequences and Series Find r when a3 = 48 and a6=3072.
II. Subtract the smaller term number from the larger term number. This will become the nth root.
III. Find the nth root of the quotient.
III. Find the nth root of the quotient.
Jan 2412:50 PM
Analyze Geometric Sequences and Series Now find a1 using one term, a3 = 48 and the r value you just calculated.
an = a1 r n1
48= a1 (4)
Jan 2412:50 PM
Analyze Geometric Sequences and Series One term of an geometric sequence is a3 = 48 and another term is a6=3072. Now substitute the values you calculated for a1 and r.
31
a1 = 3
r = 4
48= a1 (4)2
an = a1 r n1
48= a1 (16)
an = 3( 4 ) n1
a1 = 3 Jan 2412:50 PM
Analyze Geometric Sequences and Series Sum of a finite geometric series
Jan 2412:50 PM
Analyze Geometric Sequences and Series Find the sum of the geometric series
The sum of the first n terms of a finite geometric series with common ratio r ≠1 is:
Jan 2412:50 PM
Jan 2412:50 PM
3
Algebra 2 CH 7.3 Analyze Geometric Sequences and Series.notebook
Analyze Geometric Sequences and Series Find the sum of the geometric series
n=8
a1=3
February 11, 2013
Analyze Geometric Sequences and Series a1=3 r = 4 n=8
r = 4
Jan 2412:50 PM
Analyze Geometric Sequences and Series
Jan 2412:50 PM
Analyze Geometric Sequences and Series Find the sum of the geometric series
n=30
Jan 2412:50 PM
Analyze Geometric Sequences and Series n=30 a1=10 r = 0.25
Jan 2412:50 PM
a1=10
r = 0.25
Jan 2412:50 PM
Analyze Geometric Sequences and Series
Jan 2412:50 PM
4
Algebra 2 CH 7.3 Analyze Geometric Sequences and Series.notebook
Analyze Geometric Sequences and Series In 1990, the total box office revenue at US movie theaters was about $5.02 billion. From 1990 to 2003, the total box office revenue increased by about 5.9% per year.
Write a rule for the total box office revenue a n in terms of the year. Let n=1 represent 1990.
an = 5.02( 1.059 n1 )
Jan 2412:50 PM
Analyze Geometric Sequences and Series
February 11, 2013
Analyze Geometric Sequences and Series Make a prediction for the movie revenue for 2012.
an = 5.02( 1.059 n1 )
a22 = 5.02( 1.059 221 )
a22 = 5.02( 3.3328)
a22 = 16.73 billion
Jan 2412:50 PM
Analyze Geometric Sequences and Series
What was the total box office revenue at US movie theaters from 1990 to 2003?
an = 5.02( 1.059 n1 )
Jan 2412:50 PM
Analyze Geometric Sequences and Series
Jan 2910:11 AM
Analyze Geometric Sequences and Series 7.3 Analyze Geometric Sequences and Series • Geometric sequence • Common ratio • Geometric sequence
What was the total box office revenue at US movie theaters from 1990 to 2003?
Feb 107:57 PM
Jan 2412:50 PM
5
February 11, 2013
Analyze Geometric Sequences and Series
Analyze Geometric Sequences and Series
In an geometric sequence, the ratio between consecutive terms is constant.
7.3 Analyze Geometric Sequences and Series • Geometric sequence
3, 12, 48, 192, 768, 3072... x4 x4 x4 x4 x4
• Common ratio • Geometric sequence
The ratio is called the common ratio.
x
x
x
x
x
625, 125, 25, 5, 1, 1/5... Jan 2412:50 PM
Jan 2412:50 PM
,
Analyze Geometric Sequences and Series 3, 12, 48, 192, 768, 3072... The common ratio is calculated by dividing a number by the previous number.
Analyze Geometric Sequences and Series Rule for an geometric sequence: an = a1 (r n1 )
What formula that we used in the past does this look like? Jan 2412:50 PM
Analyze Geometric Sequences and Series
an = a1 r n1 Write a rule for the nth term of the sequence:
Jan 2412:50 PM
Analyze Geometric Sequences and Series
1/9, 1/3, 1, 3, 9, 27,... We need to find a1 and r.
a1 = 1/9
r = 3
1/9, 1/3, 1, 3, 9, 27,...
an =1/9 ( 3 n1) Now find the 20 th term.
Now find the 20 th term.
Jan 2412:50 PM
Jan 2412:50 PM
1
Algebra 2 CH 7.3 Analyze Geometric Sequences and Series.notebook
Analyze Geometric Sequences and Series One term of a geometric sequence is a5 = 324 and the common ratio is 6.
February 11, 2013
Analyze Geometric Sequences and Series One term of a geometric sequence is a5 = 324 and the common ratio is 6. Find the a1:
Find the rule for the nth term
an = a1 r n1
Jan 2412:50 PM
Analyze Geometric Sequences and Series
an = a1 r n1
324 = a 1(6 51 )
Jan 2412:50 PM
Analyze Geometric Sequences and Series
324 = a 1(6 51 )
One term of a geometric sequence is a5 = 324 and the common ratio is 6.
324 = a 1(6 4)
Find the rule for the nth term
324 = a 1(1296)
0.25 = a 1
a1=4
r = 6
an = a1 r n1
an =0.25 ( 6 n1) Jan 2412:50 PM
Analyze Geometric Sequences and Series
Jan 2412:50 PM
Analyze Geometric Sequences and Series
One term of an geometric sequence is a3 = 48 and another term is a6=3072.
Find r when a3 = 48 and a6=3072.
Find the rule for the nth term
I. Divide the latter term by the earlier term:
an = a1 r n1
Jan 2412:50 PM
Jan 2412:50 PM
2
Algebra 2 CH 7.3 Analyze Geometric Sequences and Series.notebook
Analyze Geometric Sequences and Series Find r when a3 = 48 and a6=3072.
February 11, 2013
Analyze Geometric Sequences and Series Find r when a3 = 48 and a6=3072.
II. Subtract the smaller term number from the larger term number. This will become the nth root.
III. Find the nth root of the quotient.
III. Find the nth root of the quotient.
Jan 2412:50 PM
Analyze Geometric Sequences and Series Now find a1 using one term, a3 = 48 and the r value you just calculated.
an = a1 r n1
48= a1 (4)
Jan 2412:50 PM
Analyze Geometric Sequences and Series One term of an geometric sequence is a3 = 48 and another term is a6=3072. Now substitute the values you calculated for a1 and r.
31
a1 = 3
r = 4
48= a1 (4)2
an = a1 r n1
48= a1 (16)
an = 3( 4 ) n1
a1 = 3 Jan 2412:50 PM
Analyze Geometric Sequences and Series Sum of a finite geometric series
Jan 2412:50 PM
Analyze Geometric Sequences and Series Find the sum of the geometric series
The sum of the first n terms of a finite geometric series with common ratio r ≠1 is:
Jan 2412:50 PM
Jan 2412:50 PM
3
Algebra 2 CH 7.3 Analyze Geometric Sequences and Series.notebook
Analyze Geometric Sequences and Series Find the sum of the geometric series
n=8
a1=3
February 11, 2013
Analyze Geometric Sequences and Series a1=3 r = 4 n=8
r = 4
Jan 2412:50 PM
Analyze Geometric Sequences and Series
Jan 2412:50 PM
Analyze Geometric Sequences and Series Find the sum of the geometric series
n=30
Jan 2412:50 PM
Analyze Geometric Sequences and Series n=30 a1=10 r = 0.25
Jan 2412:50 PM
a1=10
r = 0.25
Jan 2412:50 PM
Analyze Geometric Sequences and Series
Jan 2412:50 PM
4
Algebra 2 CH 7.3 Analyze Geometric Sequences and Series.notebook
Analyze Geometric Sequences and Series In 1990, the total box office revenue at US movie theaters was about $5.02 billion. From 1990 to 2003, the total box office revenue increased by about 5.9% per year.
Write a rule for the total box office revenue a n in terms of the year. Let n=1 represent 1990.
an = 5.02( 1.059 n1 )
Jan 2412:50 PM
Analyze Geometric Sequences and Series
February 11, 2013
Analyze Geometric Sequences and Series Make a prediction for the movie revenue for 2012.
an = 5.02( 1.059 n1 )
a22 = 5.02( 1.059 221 )
a22 = 5.02( 3.3328)
a22 = 16.73 billion
Jan 2412:50 PM
Analyze Geometric Sequences and Series
What was the total box office revenue at US movie theaters from 1990 to 2003?
an = 5.02( 1.059 n1 )
Jan 2412:50 PM
Analyze Geometric Sequences and Series
Jan 2910:11 AM
Analyze Geometric Sequences and Series 7.3 Analyze Geometric Sequences and Series • Geometric sequence • Common ratio • Geometric sequence
What was the total box office revenue at US movie theaters from 1990 to 2003?
Feb 107:57 PM
Jan 2412:50 PM
5
