1. Determine if the sequence is arithmetic. If it is, find the common difference. 35, 32, 29, 26,......
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Transcript of 1. Determine if the sequence is arithmetic. If it is, find the common difference. 35, 32, 29, 26,......
Warm up 1. Determine if the sequence is
arithmetic. If it is, find the common difference.• 35, 32, 29, 26, ...
2. Given the first term and the common difference of an arithmetic sequence find the first five terms and the explicit formula.• a1 = 28, d = 10
Lesson 12-2 Geometric Sequences & Series
Objective: To find the nth term and the geometric means of a
geometric sequenceTo find the sum of n terms of a
geometric series
Geometric sequences increase by a constant factor called the common ratio (r)• an = ran-1
Geometric Sequences are also called geometric progressions.
Geometric Sequence
Common ratio
1, 2, 4, 8, 16 … is an example of a geometric sequence with first term 1 and each subsequent term is 2 times the term preceding it. The common ratio (r), is 2
Geometric Sequences and Series
A geometric sequence is a sequence in which each term after the first is obtained by multiplying the preceding term by a constant nonzero real number.
Finding the Common Ratio
In a geometric sequence, the common ratio can be found by dividing any term by the term preceding it.
The geometric sequence 2, 8, 32, 128, …has common ratio r = 4 since8 32 128
... 42 8 32
Geometric Sequences and Series
nth Term of a Geometric Sequence
In the geometric sequence with first term a1 and common ratio r, the nth term an, is
11n
na a r
Using the Formula for the nth Term
Example Find a5 and an for the geometric
sequence 4, –12, 36, –108 , …
Solution Here a1= 4 and r = 36/ –12 = – 3. Using
n=5 in the formula
In general
5 1 45 4 ( 3) 4 ( 3) 324a
1 11 4 ( 3)n n
na a r
11n
na a r
Modeling a Population of Fruit Flies
Example A population of fruit flies grows in such a
way that each generation is 1.5 times the previous
generation. There were 100 insects in the first generation. How many are in the fourth
generation.
Solution The populations form a geometric sequence
with a1= 100 and r = 1.5 . Using n=4 in the formula
for an gives
or about 338 insects in the fourth generation.
3 34 1 100(1.5) 337.5a a r
Geometric Means
In a geometric sequence the terms between two nonconsecutive terms are called geometric means.
Practice Write a sequence that has two
geometric means between 128 and 54.
Find the common ratio.
Use r to find the other terms. 128, 96,72,54
312854 r3
128
54r
3
64
27r
3 33
64
27r
r4
3
11n
na a r
Geometric Series
A geometric series is the sum of the terms of a geometric sequence .
In the fruit fly population model with a1 = 100 and r = 1.5, the total population after four generations is a geometric series:
1 2 3 4
2 3100 100(1.5) 100(1.5) 100(1.5)
813
a a a a
Geometric Sequences and Series
Sum of the First n Terms of an Geometric Sequence
If a geometric sequence has first term a1 and common ratio r, then the sum of the first n terms is given by
or
where .
1(1 )
1
n
n
a rS
r
1r
r
raaS
n
n
1
11
Finding the Sum of the First n Terms
Example Find the sum of the first six terms of a geometric series if a1 = 6 and r=3.
Solution 6
6
6(1 3 ) 6(1 729) 6( 728)2184
1 3 2 2S
Geometric vs. arithmetic sequences
The difference is in how they grow
Arithmetic sequences increase by a constant amount• an = 3n• The sequence {an} is { 3, 6, 9, 12, … }• Each number is 3 more than the last• Of the form: f(x) = dx + a
Geometric sequences increase by a constant factor• bn = 2n
• The sequence {bn} is { 2, 4, 8, 16, 32, … }• Each number is twice the previous• Of the form: f(x) = arx
Practice Determine if the sequence is
geometric. If it is, find the common ratio.
1) −1, 6, −36, 216, ... 2) −1, 1, 4, 8, ...
Given the first term and the common ratio of a geometric sequence find the first five terms and the explicit formula.
3)a1= 0.8, r = −5 4) a1= 1, r = 2