1. Math 20-1 Chapter 1 Sequences and Series
Teacher Notes
1.3 Geometric Sequences

2. 1.3 Geometric Sequences Math 20-1 Chapter 1 Sequences and Series
Many types of sequences can be found in nature. The Fibonacci sequence, frequently found in flowers, seeds, and trees, is one example. A geometric sequence can be approximated by the orb web of the common garden spider.
The lengths of the sections of the silk between the radii for this section of the spiral produce a geometric sequence.
8, 12, 18, 27, 40.5, 60.75
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3. Geometric Sequences Examples Non-Examples
Geometric Sequences
Examples
Non-Examples
2, 10, 50 …
250
10, 15, 20, 25, …
-4, 12, -36, 108, …
-324
5, 4, 3, 2, 1, …, -102, -103
Where would these sequences go?
Geometric Sequences
What do you think makes a sequence geometric?
What is the next term in each geometric sequence?
1.3.2

4. Characteristics of a Geometric Sequence
8, 12, 18, 27, 40.5, 60.75
Is the relationship between consecutive terms of the geometric sequence different from an arithmetic sequence?
For several pairs of consecutive terms in the sequence, divide a term by the preceding term.
What did you notice?
An ordered list of terms, separated by commas, in which the ratio of consecutive terms is constant is called a geometric sequence.
Multiplying any term by a constant called the common ratio, r, will give the next term in the geometric sequence.
1.3.3

5. Consider the sequence a) What is the value of t1? 4 t4? 32
4, 8, 16, …
a) What is the value of t1? 4
t4? 32
b) Determine the value of the common ratio.
c) What strategies could you use to determine the value of t10?
d) What would the graph of the geometric sequence differ from the graph of an arithmetic sequence?
1.3.4

6. Deriving a Rule for the General Term of a Geometric Sequence
Terms
Sequence
Sequence Expressed
using first term and
common ratio
General Sequence 4 8 16 32 4
4(2)
4(2)(2)
4(2)(2)(2)
4 (2)…(2)
A geometric sequence is a sequence that has a constant
common ratio, r, between successive terms.
Position of
term in
the sequence
tn = t1 rn-1
General
term or
nth term
First
term
Common
ratio
What assumptions are made?
1.3.5

7. tn = t1 rn-1 tn = t1 rn-1 tn = 4(2)n-1 t10 = 4 (2)10-1 = 4 (2)9
4, 8, 16 …
parameters t1 and r must be defined
Determine the value of t10.
Write the expression for the general term.
tn = t1 rn-1
Explicit Definition
t1 = 4
n = var
r = 2
tn = t1 rn-1
t1 = 4
n = 10
r = 2
t10 = ?
tn = 4(2)n-1
t10 = 4 (2)10-1
= 4 (2)9
t10 = 2048
Is it mathematically correct to multiply the 4 and 2?
t10 = 4(2)10-1
t10 = 4(2)9
t10 = 2048
1.3.6

8. tn = t1 rn-1 tn = 1(2)n-1 tn = (2)n-1 t9 = (2)9-1 t9 = 256 t1 = 1
When Spider-Man was bitten, the radioactive spider injected 1 mg of venom into his body. The venom concentration doubled every hour.
Write an expression that models the concentration of venom in Spider-Man’s body over time.
tn = t1 rn-1
t1 = 1
r = 2
n = var
tn = 1(2)n-1
How many mg were in Spider-Man’s blood stream eight hours after being bitten?
tn = (2)n-1
What is the value of n? Explain.
t9 = (2)9-1
t9 = 256
There would be 256 mg of venom after eight hours.
1.3.7

9. Assignment Suggested Questions Page 39:
1d,e,f, 2a, 3c, 5a,b, 6a, 7, 8, 10, 14
1.3.8