Geometric
Series
Finite Geometric Series
A finite geometric series is the expression for the sum of the
terms of a finite geometric sequence.
2, 6,18, 54
is a geometric sequence
2  6  18  54
is a geometric series
The General formula for the Sum of the first n terms
𝑎(𝑟 𝑛 − 1)
𝑆𝑛 =
,𝑟 ≠ 1
𝑟−1
𝑟𝑎𝑛 − 𝑎
𝑆𝑛 =
,𝑟 ≠ 1
𝑟−1
Known Values are:
Known Values are:
a
r
n
a
r
an
Why is there a restriction on the value of r?
1.4.3
Calculate the sum of the series 5 + 15 + 45 + . . . + 10 935.
𝑎(𝑟 𝑛 − 1)
𝑆𝑛 =
,𝑟 ≠ 1
𝑟−1
𝑆𝑛 =
𝑟𝑎𝑛 − 𝑎
,𝑟 ≠ 1
𝑟−1
3(10935)  5
Sn 
 3  1
a1 = 5
r= 3
n= ?
an = 10 935
PEMDAS
Sn = 16 400
The sum of the finite geometric series is 16 400.
1.4.4
Finite Geometric Series
Algebraically determine the sum of the first seven terms of the series
27 + 9 + 3 + . . ..
𝑎(𝑟 𝑛 − 1)
𝑆𝑛 =
,𝑟 ≠ 1
𝑟−1
𝑟𝑎𝑛 − 𝑎
𝑆𝑛 =
,𝑟 ≠ 1
𝑟−1
 1  7 
27    1
 3 

S7 
1 
  1
3 
S7 
1093
27
a1 = 27
r = 1/3
n= 7
an = ?
The sum of the first seven terms is
1093
27
or approximately 40.5.
1.4.5
Finite Geometric Series
How many terms of the series 2 + (-4) + 8 + (-16) + . . . will yield a
sum of 342?
𝑎(𝑟 𝑛 − 1)
𝑆𝑛 =
,𝑟 ≠ 1
𝑟−1
2[( 2) n  1]
342 
 2  1
𝑆𝑛 =
𝑟𝑎𝑛 − 𝑎
,𝑟 ≠ 1
𝑟−1
a1 = 2
r = -2
n= ?
Sn = 342
an = ?
-1026 = 2[(-2)n – 1]
- 513 = (-2)n - 1
What strategy could you use to determine the value of n?
-512 = (-2)n
(-2)9 = (-2)n
9=n
For this geometric series, nine terms
must be added for a sum of 342.
1.4.6
Determining the First Term from a Series
The common ratio of a geometric series is ¼ and the sum of
the first 4 terms is 1275. What is the value of the first term?
𝑎(𝑟 𝑛 − 1)
𝑆𝑛 =
,𝑟 ≠ 1
𝑟−1
𝑟𝑎𝑛 − 𝑎
𝑆𝑛 =
,𝑟 ≠ 1
𝑟−1
 1  4 
t1    1
 4 

1275 
1 
  1
4 
a1 = ?
r = 1/4
n= 4
S4 = 1275
an = ?
256 
 1
t1 

256 256 

1275 
 3
 
 4

3825
 255 
 t1  

4
 256 
960  t1
1.4.9