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Sequences - Packet #2
Geometric Sequences
Definition: A geometric sequence is a sequence in which each term, after the first is formed by multiplying
the previous term by a fixed quantity.
A geometric sequence may be represented as
where a1 represents the first term, and r is the common ratio. The common ratio is a constant which is
multiplied by each term of a geometric sequence to produce the next term.
 In the sequence 2, 4, 8, 16, 32… the initial term (
is 2 and the common ratio
is 2
 In the sequence 1, 3, 9, 27, 81… the initial term (
is 1 and the common ratio
is 3
 In the sequence 3, 6, 12, 24… the initial term (
is 3 and the common ratio
is
This type of sequence has the property that each term is multiplied by a constant to produce the next term.
Calculating the common ratio: The common ratio is the ratio of any term in the sequence to the term
preceding it and can be found by dividing any term by the one before it. If r is the common ratio in a
geometric sequence, then
The existence of a common ratio between any two successive terms is the characteristic feature of a
geometric sequence. To test whether a given sequence is geometric, determine whether the ratio of any
given term to the immediately preceding one is always the same.
 For example, 64, 32, 8, 4… is not a geometric sequence because 32 64 = ½, but 8 32 = ¼
Exercises:
H. Write the first four terms of the geometric sequence in which a1 and r are as follows.
25.) a1 = 13, r = 7
26.) a1 = 4, r =
1
2
27.) a1 =7, r = -2
I. Write the last four terms before a n of the geometric sequence in which a n and are as follows.
28.) a n =324 r = 2
29.) a n = 639, r = 3
30.) a n = -400, r = -5
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If a1 represents the first term of a geometric sequence, an the nth term, n the term number and r the common
ratio, then the formula for finding the nth term is:
*not given on the regents exam reference sheet
Model Solutions to Common Questions using this formula
Finding nth term given a piece of the sequence
Ex: Find the 8th term of the sequence 243, 81, 27, 9, ….
 State the formula

List known & unknown values

Substitute into the formula

Solve
Identify the common ration given two non-consecutive terms
Ex: Find the common ratio and write the geometric sequence whose first term is 4 and whose 3rd
term is 36
 State the formula

List known & unknown values

Substitute into the formula


Solve
Apply the solution as a series
4, 12, 36….
Hey, look at that.. the 3rd term is 36!
Identify the position of a value
Ex: What term of the sequence 1/8, -1/4, ½, -1, 2, -4, …. is 128?

State the formula

List known & unknown values

Substitute into the formula

Solve
(either switch to logs, or modify bases in exponential form)

Apply the solution
128 is the 11th term
Exercises:
J. Find the term indicated in each of the following geometric sequence.
31.) 6th term of
1
, -1, 2, -4, …..
2
33.) 9th term of 27, 9, 3, …….
32.) 10th term of 3, 6, 12, ……..
34.) 7th term of 2, 6, 18, …….
K. Find the n’th term of the geometric sequence in which:
35.) a1 = 3, r = 3, n = 10
36.) a1 = 7, r = -4, n = 5 37.) a1 = 800, r = 1, n = 7
L. Answer each question.
38.) Which term of 7, 14, 28, ….. is 14336?
39.) What term of 2187, -729, 243, …. is 27?
40.) Find the common ratio of the geometric sequence, whose first term is 5 and whose
4th term is -320.
41.) Find the common ratio of the geometric sequence whose first term is 4 and whose 3rd
term is 36.
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Geometric means
Definition: Geometric means are the terms between any two other terms of a geometric sequence.
Model Solution
Ex: Insert 2 geometric means between 3 and 375
 Write the sequence, leaving blank spaces for the missing means
3, ____, ____, 375

Determine the values of a1, an, and n
a1 = 3, a4 = 375, n = 4

Substitute in the formula for a1, an, n and solve for r.

Write the sequence by multiplying the value of r to determine the
next term.
3, 15, 75, 375

Answer the question appropriately
The two geometric
means are 15 and 75.
Exercises:
M. In each of the following, insert the indicated number of geometric means between the two given
numbers.
42.) 1 and 27, 2 means
43.) -2 and 54, 2 means
44.) 16 and 1, 3 means
N. Answer each question as indicated for each problem.
45.) Find the positive geometric mean between 4 and 25.
46.) Find the negative geometric mean between 4 and 64.
47.) Between 32 and what other number is the positive geometric mean 8?