Transcript Document
SS.01.3 - Geometric Sequences
MCR3U
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(A) Review
A sequence is an ordered set of numbers.
An arithmetic sequence has a pattern to it =>
the constant difference between successive
terms.
Today, we will explore other sequences that
have another pattern to them.
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(B) Geometric Sequences
Comment upon any pattern you see in the sequences
- i.e. what makes these sequences easy to work
with??
ex
ex
ex
ex
ex
1.
2.
3.
4.
5.
2,10,50,250,.....
5,-10,20,-40,80,.....
6, 0.6, 0.06, 0.006, 0.0006,....
2,4,8,16,32,64,….
100, 50, 25, 12.5, 6.25, …
Each pair of successive terms have a constant ratio
=> thus making them Geometric Sequences
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(C) Representing the Geometric Sequences
(1) Table of Values
Time
Amount
0
2
1
4
2
8
3
16
4 5
32 64
from which we notice no common first difference
but if we divide each term by the preceding term, we
notice a common ratio
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(C) Representing the Geometric Sequences
(2) Scatter plots
from which we notice a curved relation
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(C) Representing the Geometric Sequences
(3) Equations and Formulas
But how do we determine the formula that
generates the terms of the sequence??
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(D) The General Term of a Geometric
Sequences
Consider the following analysis:
t1
t2
t3
t4
t5
=
=
=
=
=
3 = 3 x 1 = 3 x 20
6 = 3 x 2 = 3 x 21
12 = 3 x 4 = 3 x 2 x 2 = 3 x 22
24 = 3 x 8 = 3 x 2 x 2 x 2 = 3 x 23
48 = 3 x 16 = 3 x 2 x 2 x 2 x 2 = 3 x 24
We can see a pattern emerging as to how to calculate the
general term of a geometric sequence as: tn = arn-1, where a is
the first term of the sequence, n is the term number, and r is
the common ratio.
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(D) The General Term of an Arithmetic
Sequences
Working with the formula tn
notice two things:
= arn-1, we
If r > 1, then the terms increase
If 0 < r < 1, then the terms decrease
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(E) Examples
ex 1.
Write the first 6 terms of the sequence defined by tn = 5(-2)n-1
ex 2.
term.
Given the formula for the nth term as tn = -5(4)n-1, find 10th
ex 3. Find the formula for the nth term given the geometric sequence
2,6,18,...... Then find the 7th term.
ex 4. How many terms are there in the geometric sequence
3,6,12,....,384
ex 5. If the 5th term of a sequence is 1875 and the 7th term is
46,875, find a, r, and tn and the first three terms of the sequence.
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Examples
ex 6. Since 1967, the average annual baseball
salary was $19,000. The average annual salary has
been rising at a rate of 17% per year. Determine the
equation for geometric sequence and then predict
the average annual salary for 2007.
ex 7. The half life of iodine-131 is 8 days. What
amount will remain in 112 days if you started with 12
mg of iodine-131? Determine the equation for
geometric sequence.
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(F) Internet Links
College Algebra from WTAMU
(http://www.wtamu.edu/academic/anns/mps/m
ath/mathlab/col_algebra/col_alg_tut54c_arit
h.htm)
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Homework
AW textbook page 22-25
Q1-3,7,10,11,12,14,17,20,21,22
Nelson Text, p57-60, Q1-3,58,10,11,15,20
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