**Document ID**: 01_30_00_1

**Date Received**: 2000-01-30 **Date Revised**: 2000-02-27 **Date
Accepted**: 2000-02-27

**Curriculum Topic Benchmarks**: M2.4.2, M3.4.3, M3.4.4, M3.4.5

**Grade Level**: \[9-12\] High School

**Subject Keywords**: geometric progression, counter-intuitive result,
counting, exponents, estimates

**Rating**: moderate

## Dream Job

**By**: Phyllis A Salmons, Embry-Riddle Aeronautical University, 461
Brentwood Drive, Daytona Beach FL 32117

e-mail: phyllis.salmons@msfc.nasa.gov; salmonsp@cts.db.erau.edu

**From**: The PUMAS Collection http://pumas.jpl.nasa.gov

©2000, California Institute of Technology. ALL RIGHTS RESERVED. Based on
U.S. Gov\'t sponsored research.

Many people dream of becoming millionaires these days. In order to make
this dream come true, people play the stock market, play the lottery,
and appear on television game shows designed for this purpose. Counting
on the stock market requires placing money at risk and playing the
lottery involves playing against the odds. And honestly, the television
programs place contestants in the risky position of looking foolish by
missing an easy question in front of millions of viewers.

Let us examine a plan for earning a million dollars based on a contract
between an employee and an employer. After all, this way sounds like a
respectable method \-- working for your wages. First let us agree upon a
contract.

+-----------------------------------------------------------------------+
| ***Contract for Employment***                                         |
|                                                                       |
| Employee \_\_\_\_\_\_\_\_\_\_\_\_(enter your name)                    |
|                                                                       |
| Employer \_\_\_\_\_\_\_\_\_\_\_\_(a company agreeing with these       |
| terms)                                                                |
|                                                                       |
| ***Points of Agreement***                                             |
|                                                                       |
| 1\. The employee will work a five-day work week.                      |
|                                                                       |
| 2\. The employee will be paid for the week's wages each Friday.       |
|                                                                       |
| 3\. The employee will be hired for a minimum of 30 work days.         |
|                                                                       |
| 4\. The salary schedule is as follows:                                |
|                                                                       |
| -   The base pay for Day 1 is one penny.                              |
|                                                                       |
| -   Each subsequent day, the salary is double that of the previous    |
|     day.                                                              |
|                                                                       |
| Signed                                                                |
| \_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_(Employee)      |
|                                                                       |
| Sig                                                                   |
| ned\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_(Employer) |
|                                                                       |
| Date: \_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_                  |
+-----------------------------------------------------------------------+

Is this a good deal? It certainly looks as if this employee has made a
bad deal. Just to test our expectations, guess how much money this
employee will have earned in the 30 working days:

My Guess \$\_\_\_\_\_\_\_\_\_\_\_. Calculate the amount one would earn
working six weeks (40-hours a week) at minimum wage? Minimum wage salary
(before taxes and other deductions) \$\_\_\_\_\_\_\_\_. Now let's
calculate the earnings for this contract and see whether the employer or
the employee has made the better deal.

**Week 1**

  ------------------------- ---------------------- ----------------------
  **Day**                   **Salary**             **Earned**

                            **Dollars**            **Cents**

  Day 1 \-- Monday          0                      1

  Day 2 \-- Tuesday         0                      2

  Day 3 \-- Wednesday       0                      4

  Day 4 \-- Thursday        0                      8

  Day 5 \-- Friday          0                      16

                                                   

  **Paycheck for week one** 0                      31

  **Total wages earned**    0                      31
  ------------------------- ---------------------- ----------------------

What do you think? You worked hard all week. You spend the weekend
soaking your feet and resting your bones. Maybe you made a bad deal \--
you are under contract for 30 days. Chin up \-- only 25 more days to go!

**Week Two**

  ------------------------- ---------------------- ----------------------
  **Day**                   **Salary**             **Earned**

                            **Dollars**            **Cents**

  Day 6 \-- Monday          0                      32

  Day 7 \-- Tuesday         0                      64

  Day 8 \-- Wednesday       1                      28

  Day 9 \-- Thursday        2                      56

  Day 10 \-- Friday         5                      12

                                                   

  **Paycheck for week two** 9                      92

  **Total wages earned**    10                     23
  ------------------------- ---------------------- ----------------------

After two exhausting weeks you have only earned a total of \$10.23!
Certainly your employer is laughing behind your back!

**Week Three**

  ------------------------- ---------------------- ----------------------
  **Day**                   **Salary**             **Earned**

                            **Dollars**            **Cents**

  Day 11 \-- Monday         10                     24

  Day 12 \-- Tuesday        20                     48

  Day 13 \-- Wednesday      40                     96

  Day 14 \-- Thursday       81                     92

  Day 15 \-- Friday         163                    84

                                                   

  **Paycheck for week       317                    44
  three**                                          

  **Total wages earned**    327                    67
  ------------------------- ---------------------- ----------------------

The weekly wage is getting better, but still, you have worked three
whole weeks for only \$327.67. What a shame you made such an agreement!

**Week Four**

  ------------------------- ---------------------- ----------------------
  **Day**                   **Salary**             **Earned**

                            **Dollars**            **Cents**

  Day 16 \-- Monday         327                    68

  Day 17 \-- Tuesday        655                    36

  Day 18 \-- Wednesday      1310                   72

  Day 19 \-- Thursday       2621                   44

  Day 20 \-- Friday         5242                   88

                                                   

  **Paycheck for week       10,158                 08
  four**                                           

  **Total wages earned**    10,485                 75
  ------------------------- ---------------------- ----------------------

Oh my! This is interesting! Where can I get an application for
employment? By the way, I heard that your employer has been in touch
with the corporate law firm about the consequences of breaking a
contract. And just think \-- you still have to work for 10 more days!

**Week Five**

  ------------------------ ----------------------- -----------------------
  **Day**                  **Salary**              **Earned**

                           **Dollars**             **Cents**

  Day 21 \-- Monday        10,485                  76

  Day 22 \-- Tuesday       20,971                  52

  Day 23 \-- Wednesday     41,943                  04

  Day 24 \-- Thursday      83,886                  08

  Day 25 \-- Friday        167,772                 16

                                                   

  **Paycheck for week      325,058                 46
  five**                                           

  **Total wages earned**   335,544                 31
  ------------------------ ----------------------- -----------------------

You must be a genius! How did you find this job? Do they have any
openings? Can I apply for the job when your contract is complete? You
only have one more week to go!

**Week Six**

  ------------------------ ----------------------- -----------------------
  **Day**                  **Salary**              **Earned**

                           **Dollars**             **Cents**

  Day 26 \-- Monday        335,544                 32

  Day 27 \-- Tuesday       671.088                 64

  Day 28 \-- Wednesday     1,342,177               28

  Day 29 \-- Thursday      2,684,354               56

  Day 30 \-- Friday        5,368,709               12

                                                   

  **Paycheck for week      10,401,873              92
  six**                                            

  **Total wages earned**   10,737,418              23
  ------------------------ ----------------------- -----------------------

You became a millionaire on day \# 27 (total salary) and made over \$1
million on day 28 alone! By the end of 30 days, you have accumulated
over \$10 million!

**Background**

Do you think there is a formula that might allow one to calculate a
particular day's salary without having to calculate every step? Of
course there is! The dream job calculation is a classic example of a
**geometric progression**.

A geometric progression is a sequence of numbers in which the ratio of
any number to the number before it is a constant amount, called the
**common ratio**. For example, the sequence of numbers 1, 2, 4, 8, 16,
\... has a common ratio of 2. A geometric progression may be described
by calling the first term in the progression X (in our case X is one
cent), the common ratio as R (in our case, R=2), and in a finite
progression, the number of terms as n. The nth term of a geometric
progression is given by: X~n~ = X~1~R^n-1^

Obviously this problem could have been presented using other setups (not
necessarily money), as long as the geometric progression is shown. A
legendary treatment of the problem (according to a source at
http://www.np.ac.sg/\~bms.SereBino/19-2GePr.htm) involves a king who
promised a prince anything he wanted because the prince saved the
princess's life. The prince requested one grain of rice on the first
square of a chessboard, 2 on the 2nd, 4 on the 3rd, etc. Check the web
site to see how much rice (by weight) the king would eventually need to
place on the chessboard to satisfy this agreement.

Another source discusses it just the way we did, starting with one penny
and doubling the amount everyday. You may see that treatment at
"Doubling Pennies":

http://forum.swarthmore.edu/dr.math/problems/valerie11.26.96.html.

The key for clever math students is realizing how rapidly the sum grows
and understanding the geometric progression formula(s). Hope you enjoyed
it!

**Dream Job Questions**

1\. How does the total amount compare with your original guess?

2\. Suppose you wanted to buy a car. On which day could you purchase
your dream car and pay in cash?

3\. This calculation does not take into account income taxes! Also,
contributions must be made to Social Security and to Medicare! Look up
the tax tables on the web and determine how much income tax and other
contributions you will need to pay!

4\. What percentage of the total salary (before taxes and withholding)
was earned in week 1? In week 6?

5\. Can you state a formula for the daily salary?

**Answer:** Daily Salary = 2^n-1^ X

where n = the number of days you've been working and X = your base
salary on day one

6\. Can you state a formula for the accumulated salary on a given day?

**Answer:** Accumulated Salary = 2^n-1^ X (from n = 1 to N)

where n = the number of days you've been working; N = the number of
given day for the calculation and X = your base salary on day one

7\. Did you notice anything special about the relationship between
Friday's paycheck of one week and Monday's paycheck of the next week?
State a formula giving this connection.

**Answer:** 2^n^ X (from n = 0 to N) = (2^n+1^-1 )X

where n = the number of days you've been working; N = the number of
given day for the calculation and X = your base salary on day one

8\. Write a computer program to tell you daily salary and accumulated
salary on a given day.

9\. Using the geometric series described by 1, 3, 9, 27,\... determine
the 8th term. What is the formula for the nth term?

**Answer:** 2187; X~N~ = 3^n-1^ (from n = 0 to N)

**Application of this counting principal to a social cause**

There are many worthy social issues that deserve the attention of
politicians and the general public. You may feel very strongly about
discouraging the use of illegal drugs or about the treatment of
wildlife. No matter how strongly you feel about any of the countless
issues in our world today, you may also feel overwhelmed and discouraged
about the impact of one lone person. The above illustration shows that
it is not impossible for one person committed to a cause to make a
difference. Let's think about this:

What can you do to raise social consciousness about this very important
issue? You have no money to launch a major campaign. You have no
provision to make a television plea or to reach a large number of people
in the newspapers or magazines. Social issues, however, are often
affected by "grass-roots" efforts \-- with it all started by a scant few
individuals who are committed to a cause.

Suppose you tell one person a day about your issue. One on one you can
make a much more convincing argument than you could in a 15-second
television spot. A one-on-one plea will be much more effective in
convincing the listener.

One the second day, there will be two of you who can approach two more
people. On day three, there are four of you to approach four more
people. On day five, the eight of you convince eight more people and so
forth. By day 12, there are over 2,000 people who know about your cause
and who are each taking the time every day to tell just one more person
about the cause! On day 30, over one billion people are talking about
the issue that is so close to your heart!

You personally talked to 30 people. YOU CAN DO THAT! You have impacted a
billion people! Imagine that. Even if only a fraction of the contacts is
truly successful in convincing the listener, still you can make a
difference!

**Social Impact Questions**

1\. What are some successful grass-roots efforts that have resulted in
increased awareness or social reform?

2\. If you could successfully perform the social campaign discussed
above, how many days would it take to reach the population of your
state?

3\. Suppose you win an award and you tell one new person an hour.
Everyone who knows the news also tells one new person an hour. How long
would it take everyone in your class to hear your news? How long before
everyone in your state heard the news? Your state? The United States?
The World? Do you understand how gossip or unfounded rumors spread so
rapidly?

**Conclusion**

Let's face it \-- none of us are likely to get the dream job described
above, but we can make a difference in the world when it comes to issues
we really care about \-- one person at a time. Let your friends and
family know about the issues that are important to you! You *can* change
the world! You *can* make a difference!