August 2001 Question:  "The deal of the century!" chuckled the MLB team scout to himself as he hung up the phone. "The fans and media will never believe the deal I just agreed to."
    The scout could hardly contain his excitement as he rushed to share the news with the team's owner.
    "Listen to this deal," he shouted, rushing into the owner's office. "I just got off the phone with that young rookie pitcher. You know, the one that all the teams have their eye on? Well, he may be good at pitching a ball, but he sure lacks the math skills to make a deal. Get this -- he agreed to play for $1 for the first game, $2 for the second game, $4 for the third game, and so on. As long as we double his pay each game, he'll play the entire season for us! I told him I thought this was silly -- playing for peanuts, really pennies. But he said he wasn't playing for the money. He plays because he loves the game."
    "Hmmm," the owner thought for a minute. "Are you sure you told him everything? I mean, did you explain that there are 100 games, but he would only be playing ¼ of the games, at the most? Does he understand this?"
    "Yes! Yes! I told him all that. And he still wants to play for mere pennies," replied the scout.
    "It seems too good to be true, but let's get that kid in here and sign the deal before he changes his mind!"

This pitcher must really love the game to play for such a small amount! Find out how much his total pay for the season would be if he plays ¼ of the 100 games.

Do you think the pitcher is crazy for agreeing to such a deal?


Answer:  This problem examines exponential growth an dhow using exponents quickly increases the magnitude of a number.

The pitcher is both a good pitcher and a terrific mathematician! He will be making great money this season. It may be helpful to organize the data into a chart.

Game #

Pay for game

Total for season

1

$1

$1

2

$2

$3

3

$4

$7

4

$8

$15

5

$16

$31

6

$32

$63

7

$64

$127

8

$128

$255

9

$256

$511

10

$512

$1,023

11

$1,024

$2,047

12

$2,048

$4,095

13

$4.096

$8,191

14

$8,192

$16,383

15

$16,384

$32,767

16

$32,768

$65,535

17

$65,536

$131,071

18

$131,072

$262,143

19

$262,144

$524,287

20

$524,288

$1,048,575

21

$1,048,576

$2,097,151

22

$2,097,152

$4,192,303

23

$4,194,304

$8,388,607

24

$8,388,608

$16,777,215

25

$16,777,216

$33,554,431

For pitching 25 games, this rookie pitcher would make $33,554,431!!! That's quite a deal -- for him!

There are also several interesting patterns. One is that the pay he makes for each game can be found by solving 2 (n-1) or 2 to the power of (n-1). N is the game number. For example, the 5th game would be 2 (5-1) or 2(4) or $16.