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11.3
Geometric Sequences and Series
Geometric Sequence
The ratio of any term to
the previous term is
constant.
The ratio is called the
common ratio, r.
625, 125, 25, 5, 1, …
625r = 125
(a) What is the next term in the sequence?
125r = 25
Rule for a Geometric Sequence:
(b) Is it a geometric sequence?
If so, what is the common ratio?
1, 3, 9, 27, 81, …
Infinite geometric
sequence with common
ratio,
1
r=
5
40, 28, 18, 10, 4, …
The nth term of a geometric sequence
is given by:
1st term in the
sequence
an = a1r n −1
Common Ratio
1
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Write a rule for the nth term of the
geometric sequence. Then find a7 .
Write a rule for the nth term of the
geometric sequence.
2,
152, -76, 38, -19, …
3 9 27
, ,
, ...
2 8 32
3
an = 2 
4
a2 = 15, r =
n −1
Write a rule for the nth term of the geometric sequence .
5, − 14, 39.2, − 109.76, ...
 14 
an = 5 − 
 5
a4 = 12, r = 2
1
an = 30 
2
1
2
n −1
Write a rule for the nth term of the
geometric sequence.
a3 = −48, a6 = 3072
a1 = 1, a5 = 625
n −1
2
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Write a rule for the nth term of the geometric sequence. Then find
1) 3, 15, 75, 375, ... 2) a6 = −96, r = 2
a8
Sum of a Finite Geometric Series
1− rn 

sn = a1 
−
r
1


3) a2 = −12, a4 = −3
Challenge:
What is the sum of the geometric series?
16
k −1
1. ∑ 4(3)
k =1
Find the sum of the first
10 terms given the
geometric rule:
Find x so that x, x+2 and x+3 are terms of
a geometric sequence.
n −1
2. an = 2(3)
X = -4
Now try it with your calculator ☺
3
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Sum of an Infinite Geometric Series
1
4 
∑
k =1  3 
∞
 a 
s =  1 ,
1− r 
r <1
k −1
2+
4 8
+ + ...
3 9
4