Precalculus
Notes on Geometric Series
HW: Pg 683 #33, 36, 37-45 odd, 58
Name: ________________________________________
9/26/13
GOAL: Students will calculate the sum of both finite and infinite geometric series
and apply theory to real world situations.
I. Definition and Examples
A. A geometric series is the sum of the terms of a geometric sequence.
B. Example: The originator of a chain letter writes 5 letters instructing each recipient to write 5 similar
letters to additional people. Then these people each send 5 similar letters to other people. Determine
the number of people who receive letters if the chain continues unbroken for 12 steps.
1. Does this scenario represent a geometric sequence? Explain.
2. Find the total number of people who receive letters:
Solution
General Proof
S n  5  5(5)  5(5)  5(5)  ...  5(5)
S n  a1  a1 (r )  a1 (r ) 2  a1 (r )3  ...  a1 (r ) n 1
5S n  5(5)  5(5) 2  5(5)3  ...  5(5)12
rS n  a1 (r )  a1 (r ) 2  a1 (r )3  ...  a1 (r ) n


S n  5S n  5  0  0  0  ...  5(5)12
S n  rS n  a1  0  0  0  ...  a1 (r ) n
S n (1  5)  5(1  512 )
S n (1  r )  a1 (1  r n )
2
Sn 
3
11
5(1  512 )
 305,175, 780
1 5
Sn 
a1 (1  r n )
1 r
**n is the total number of terms you have
II. Finding the sums of Finite Geometric Series
a1  2  31  6
15
A. Find the indicated sum:
 23
k
S6 
k 1
23
B. Find the indicated sum:
 1
 8   2 
n5
n
6(1  315 )
 43046718
1 3
a1  8(0.5)5  0.25
S19 
0.25(1  (0.5)19 )
 0.167
1  0.5
C. The winner of a contest will receive $1280 the first year with a 25% increase each year. What is the
total amount of money they winner will receive after 9 years?
r  1.25
a1  1280
1280(1  1.259 )
S9 
 33, 026.97
1  1.25
III. Finding the sums of Infinite Geometric Series
A. If |r|<1, then the infinite geometric sequence 𝑎1 , 𝑎1 𝑟, 𝑎1 𝑟 2 , 𝑎1 𝑟 3 , … , 𝑎1 𝑟 𝑛−1 , … has the sum:

S   a1r n 1 
n 1
a1
1 r
B. Example: Find the sum of the following infinite geometric sequence:
4, 4(0.6), 4(0.6)2 , 4(0.6)3 , … ,4(0.6)𝑛−1 , …
** Since r = 0.6 which is between -1 and 1, we can use the formula to sum infinite geometric sequences.

S   4(0.6) n 1 
n 1
4
4

 10
1  0.6 0.4
C. Example: For each, determine if the sum diverges or converges. Find the sum if it converges.
1. 1  3  9  27  ...
2. 1 
Diverges because |r|>1
1 1 1
 
 ...
3 9 27
Converges because |r|<1

1

 
n 1  3 
3.
24 12  6  3  ...
n 1

1
1
1
3

1 3

2 2
3
Converges because |r|<1

 1
24   

 2
n 1
n 1

24
24

 48
1  0.5 0.5
ADDITIONAL EXAMPLES - DAY 1
COMPOUND INTEREST
A principal of $5000 is invested in an account that earns 6% interest find the value
of the account after 18 years if the interest is compounded semiannually.
Remember:
 r
A  P 1  
 n
A = amount at the end of time
P = principal
r = annual interest rate
n = # times/year compounded
t = # years invested
nt
 0.06 
a18  5000 1 

2 

In "geometric sequence talk"
 What is the common ratio?
 Is this a geometric sequence or series problem?
 Are we finding a sum?
( 2 18)
 $14, 491.39
DEPRECIATION
A new car today will depreciate in value at an average rate of 22% per year. That
means that it retains 78% of its value. Find the depreciated value of a $34,000
new car after 5 full years.
a5  34000  0.78 
(51)
 $12, 585.12
JOB OFFERS
You accept a job offer that pays a salary of $36,000 for the first year. Suppose that
for the next 29 years you are guaranteed a 2% raise each year. How much will you
make as you begin your 30th year on the job? What would be your total
compensation over the 30 year period?
a30  (36, 000)(1.02)29  $63, 930.41
(36, 000)(1  1.0230 )
S30 
 $1, 460, 450.85
1  1.02