462
8.2. GEOMETRIC SERIES
An alternate formula for the nth partial sum is
S n = a
1 − r n
1 − r
.
Whenever |r | < 1, the infinite geometric series
∞
k=0 ar k has the finite sum
a
1−r .
Exercises
1. There is an old question that is often used to introduce the power of geometric growth.
Here is one version. Suppose you are hired for a one month (30 days, working every
day) job and are given two options to be paid.
Option 1. You can be paid $500 per day or
Option 2. You can be paid 1 cent the first day, 2 cents the second day, 4 cents the third
day, 8 cents the fourth day, and so on, doubling the amount you are paid each
day.
(a) How much will you be paid for the job in total under Option 1?
(b) Complete Table 8.3 to determine the pay you will receive under Option 2 for
the first 10 days.
Day Pay on this day Total amount paid to date
1
$0.01
$0.01
2
$0.02
$0.03
3
4
5
6
7
8
9
10
Table 8.3: Option 2 payments
8.2. GEOMETRIC SERIES
An alternate formula for the nth partial sum is
S n = a
1 − r n
1 − r
.
Whenever |r | < 1, the infinite geometric series
∞
k=0 ar k has the finite sum
a
1−r .
Exercises
1. There is an old question that is often used to introduce the power of geometric growth.
Here is one version. Suppose you are hired for a one month (30 days, working every
day) job and are given two options to be paid.
Option 1. You can be paid $500 per day or
Option 2. You can be paid 1 cent the first day, 2 cents the second day, 4 cents the third
day, 8 cents the fourth day, and so on, doubling the amount you are paid each
day.
(a) How much will you be paid for the job in total under Option 1?
(b) Complete Table 8.3 to determine the pay you will receive under Option 2 for
the first 10 days.
Day Pay on this day Total amount paid to date
1
$0.01
$0.01
2
$0.02
$0.03
3
4
5
6
7
8
9
10
Table 8.3: Option 2 payments