CHAPTER 10
LESSON 3
Teacher’s Guide
Geometric Series
AW 2.8
MP 6.3
Objective:
• To derive and apply expressions representing sums for geometric growth and to
solve problems involving geometric series
Definition:
A geometric series is the
sum
of the terms of a geometric sequence
Consider the geometric sequence 1, 2, 4, 8, 16, 32, 64, 128, 256, 512, 1024 .
If we add the terms of the sequence, we can write the geometric series as
S = 1 + 2 + 4 + 8 +16 + 32 + 64 +128 + 256 + 512 +1024 .
Instead of adding the terms directly, let’s evaluate S as follows.
We will multiply each term of the series by 2 (which is the value of the common ratio r),
realign the result under the original series, and subtract equation (2) from equation (1).
S = 1 + 2 + 4 + 8 + 16 + 32 + 64 + 128 + 256 + 512 + 1024 ....................(1)
2S =
2 + 4 + 8 + 16 + 32 + 64 + 128 + 256 + 512 + 1024 + 2048 .........(2)
S − 2S = 1− 2048
− S = −2047
S = 2047
.
We can use this process to develop a general formula for Sn , the sum of the first n terms
of a geometric series.
In general, a geometric series can be written as follows, with rSn written underneath.
Sn = a + ar + ar 2 + ar 3 + ... + ar n -2 + ar n -1 ..................(1)
ar + ar 2 + ar 3 + ... + ar n -2 + ar n -1 + ar n .........(2)
rSn =
Sn − rSn = a − ar n
Factoring Sn , we have
Sn (1− r) = a − ar n
a − ar n a(1− r n )
Finally, Sn =
=
1− r
1− r
where r ≠ 1
Alternatively, since tn = ar n-1 we can write Sn as follows:
a( 1− r n )
Sn =
(1− r )
a − ar n a − ar n -1 r a − rt n a − rl
=
=
=
1− r
1− r
1− r
1− r
where l is the last term of the geometric series (i.e., t n or ar n −1 )
=
Summary:
The sum of the first n terms of a geometric series is given by:
a(1− r n ) 
Sn =
1− r 

or
 where a is the first term, l is the last term, and r ≠ 1.

a
−
lr

Sn =

1− r
Example 1:
Determine the sum of 14 terms of the geometric series: S = 6 + 18 + 54 + ...
a(1 − r n )
1− r
6(1− 314 )
=
= 14348904
1− 3
Sn =
G–208
Example 2:
Determine the n th term, and the sum of the first n terms of the geometric sequence which
has 2, 6, and 18 as its first three numbers.
t n = ar n-1
t n = 2(3)n-1
Sn =
a(1 - r n )
(1 - r)
Sn =
2(1 - 3 n )
= 3n - 1
(1 - 3)
Example 3:
G–208
For the geometric series 1− 1 + 1 − 1 + ..., find the sum of the first 20 terms.
3 9 27
a(1− r n )
Sn =
(1− r)
  1 20 
1 1− −

20
  3  3   1  ⋅ 3
S20 =
=  1−
=
  1 
4   3 
4
 1−  −  
3 

Note to teachers: To 20 terms, this series approximates closely the sum of the infinite
a
1
series S∞ = 1− 1 + 1 − 1 + ... =
=
=3
3 9 27
1− r 1−  − 1 4
 3
Example 4:
Exam Specs, p. 10, #3
While training for a race, a runner increases her distance by 10% each day. If she runs
2 km on the first day, what will be her total distance for 26 days of training? (accurate to
2 decimal places)
a=2 

2
3
24
25
r = 1.1 S26 = 2 + 2(1.1) + 2(1.1) + 2(1.1) + ... + 2(1.1) + 2(1.1)

n = 26 
26
n
a(1− r ) 2(1− 1.1 )
=
= 218.36 km
S26 =
1− r
1 Ğ1.1
G–209
Example 5:
The following is a school trip telephoning tree.
Level 1: Teacher
Level 2: Students
Level 3: Students
A) At what level are 64 students contacted?
a=1
r = 2; (Each student contacts 2 students.)
t n = 64
Find n .
t n = ar n − 1
64 = 1⋅ 2 n −1
2 n −1 = 64
n − 1= 6
n =7
B) How many are contacted at the 8th level?
64 × 2 = 128 students
C) By the 8th level, how many students, in total, have been contacted?
Remember that the teacher is not counted. So we will subtract 1 from the total S8 .
S = (1+ 2 + 4 + 8 + 16 + 32 + 64 + 128) − 1
a( 1− r n )
Sn =
( 1− r )
1( 1− 2 8 )
= 255
( 1- 2 )
The total number of students contacted is 255 − 1 = 254 .
S8 =
D) By the n th level, how many students, in total, have been contacted?
Again, remember that the teacher is not counted. So we will simply subtract 1 from the
sum up to level n.
2( 1− 2 n )
= −2(1− 2n )
1 Ğ2
The total number of students by level n is given by
−2(1− 2n )− 1 = −2 + 2 n +1 − 1 = 2 n +1 − 3
Sn =
E) If there are 300 students in total, by what level will all have been contacted?
Since 254 students have been contacted by level 8, then all 300 will have been
telephoned by level 9.
Example 6:
1
...
What is the sum of the geometric series 1
4 + 2 + 1 + 2 + 4 + + 1024 ?
In this case, we have a = 1 and r = 2 and l = 1024
4
We can use the formula Sn =
1 − 1024(2)
4
S13 =
= 2047.75
1− 2
a − lr
1− r