G/ Accelerated Math 3
Geometric Sequences & Series
Name: ________________________
Date: _________________________
Standards: MA3A9. Students will use sequences and series
a. Use and find recursive and explicit formulae for the terms of sequences.
b. Recognize and use simple arithmetic and geometric sequences.
e. Find and apply the sums of finite and, where appropriate, infinite arithmetic and
geometric series.
f. Use summation notation to explore series.
A sequence is geometric if the ratios of consecutive terms are the same. This ratio (r) is
called the common ratio. In geometric sequences, the common ratio can be found by dividing
any term by the previous term.
For example: 4, -8, 16, -32, …
To be able to identify a sequence as geometric, check to be sure that there is a common ratio
between all given terms.
Are the following sequences arithmetic, geometric, or neither?
1.
1/32, 1/16, 1/8, 1/4, 1/2
2.
-7, -2, -8, -13, -52
3.
9, -1, -11, -21, -31, …
Explicit Formula for a Geometric Sequence
n1
n
1
a  a (r)
Ex 1: Given a1 = 6 and r = -1/4, write the explicit formula for the geometric sequence and list
the first 5 terms.

Ex 2: Write the explicit formula for the nth term of a geometric sequence. Then find the 12th
term.
4, -8, 16, -32, …
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Ex 3: Write the explicit formula for the nth term of a geometric sequence. Then find the 10th
term.
1/5, 1/25, 1/125, 1/625, …
Ex 4: Write the first five terms of the geometric sequence. Determine the common ratio and
write the nth term of the sequence as a function of n.
a1 = 5
an = -2an-1
Ex 5: Find the nth term of the geometric sequence if the first term is 32 and the fifth term is 2.
Ex 6: Find the fifteenth term of the geometric sequence with a third term of 5/4 and a sixth
term of 5/32.
Ex 7: Two terms of a geometric sequence are a2 = 45 and a5 = -1215. Find a explicit formula.
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G/ Accelerated Math 3
Geometric Sequences & Series
Name: ________________________
Date: _________________________
Sum of a Finite Geometric Series
The sum, Sn, of the first n terms of a geometric series with common ratio r is:
 1 rn
Sn  a1 
 1 r

, r  1

1 + 8 + 64 + 512 + …
Ex 1: Find the sum of the first 9 terms of the geometric series:
1
32  

4
i 1
6
Ex 2:
i 1
8
Ex 3:
 500 1.04 
n 1
n 1
Sum of an Infinite Geometric Series
The sum of an infinite geometric series with first term a1 and common ratio r is given by:
S 
a1
1 r
provided r  1 . If r  1 , the series has no sum.
Determine whether the infinite geometric series has a sum.

Ex 4:
2

3

 
3
i 1
i 1

Ex 5:
15

 
i 1 2  3 
i 1
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Find the sum of the infinite geometric series.
4
4
1
Ex 6: 12, 4, 3 , 9 , …
Ex 8: -30, 15,
1
1
Ex 7: -2, 2 , 8 , 32 , …
15 15
2 , 4 ,…
Find the sum of the infinite geometric series if it exists.

Ex 9:
 2  0.1

Ex 10:
i 1

Ex 11:
i 1
7
3

 
i 1  2 
1
10

 
2
i 0
i
i 1
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