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
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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




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=
=
=
=
=
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
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ex 2.
term.
Given the formula for the nth term as tn = -5(4)n-1, find 10th
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ex 3. Find the formula for the nth term given the geometric sequence
2,6,18,...... Then find the 7th term.
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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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