Activity 11.2 Geometric Series
Overview:
In this activity, students are given an application problem in which the number of sandwiches sold
each month increases by 20%. They are asked to complete a table that lists the number of
sandwiches sold during successive months. They are then told that there is a formula for finding
the number sold for any given month and are given the space to take notes while the instructor
discusses the formula for the sum of a geometric series.
They are then given a new scenario for which they must find the total number sold during a year.
Then they are given a scenario in which the number sold is decreasing and asked to find the
sum.
Finally, students are asked if specific sums represent a geometric series and asked to write those
that are geometric using sigma notation.
Estimated Time Required:
This activity should take approximately 20 minutes.
Technology: None
Concepts:


Geometric Series
Summation notation
Discussion:
Infinite geometric series is a difficult concept for many students. Explain that you cannot really
sum infinitely many terms. However, you can look at the sum for very large values of n, then
investigate what happens as n gets even larger.
the following is a summary of some facts about geometric series:
1. A geometric series is a sum of the form Sn  a  ar  ar 2 
 ar n 1 .
2. Any finite geometric series can be summed using the formula Sn 
ar n  a
. If n is small,
r 1
it may be more efficient to just add up the few terms directly.
3. An infinite geometric series is a sum of the form a  ar  ar 2 
4. If | r | > 1, then a  ar  ar 2 
added. It does not converge.
.
gets larger and larger without bound as more terms are
5. If | r | < 1, then the magnitude of each subsequent term being added is decreasing
a
exponentially. The result is that the partial sums converge to S 
1 r
Activity 11.2 Geometric Series
1. A fast food restaurant introduces a new sandwich. In the first month, the restaurant sells
50,000 sandwiches. The number of sandwiches sold each month increases by 20%.
Assuming this trend continues for one year, how many sandwiches does the restaurant sell
during the year?
a.) First fill in the table:
Number of
Number of sandwiches sold by the end of the nth month
Months
1
2
3
4
5
6
7
8
9
10
11
12
b.) Wouldn’t it be nice if there were a formula that would allow us to do this calculation more
efficiently? Of course there is. Take notes here:
c.) Now, if the initial number of sandwiches sold had been 40,000 and the increase had been
15% each month, how many sandwiches would have been sold the first year?
d.) Now, what if the sandwich wasn’t very popular, and even though 50,000 are sold the first
month, the number sold in each subsequent month goes down by 20%. If this trend goes on
forever, how many of the sandwiches will be sold?
4. Consider the following sums. Determine whether or not they are geometric, and if so, identify
k
a and r, and write the summation using standard sigma notation
 ar
n 1
.
n 1
a.)
1 2  3 
c.)
1 1 1
 
2 4 8
10
b.) 3  3  3 
2
d.)
3
4
 318
2  4  8 16  32