Last weekend I played eighteen holes of golf with Tiger Woods.
We bet $1 on the first hole, and doubled the bet on each succeeding hole.
He got lucky and beat me on the first seventeen holes, but I won the last hole.
(There were no ties on any hole.).
How much did I win or lose?
Solution:
The answer is that I won one dollar.
The formula for the sum of a finite geometric progression tells us that 20 + 21 + 22 + ... 2(n-1) = 2n -1.
In other words, if we start with 20 = 1, the sum of any number of consecutive terms of the geometric progression with common ratio 2 always equals one less than the next power of 2. Thus after being deeply in debt, I wound up winning $1.
(This problem illustrates the Martingale Betting System. Wikipedia has a good discussion of it.)
Here is the formula for the sum of a geometric series with the first term a, common ratio r, and n number of terms:
So, Tiger Woods' winnings would be:
My winnings would be:
My winnings = 217 = $131,072
Correctly solved by:
| 1. K. Sengupta | Calcutta, India |
| 2. Kamal Lohia |
Holy Angel School, Hisar, Haryana, India |
| 3. Davit Banana | Istanbul, Turkey |
| 4. Dr. Hari Kishan |
D.N. College, Meerut, Uttar Pradesh, India |
| 5. Seth Cohen | Concord, New Hampshire, USA |
| 6. Ivy Joseph | Pune, Maharashtra, India |
| 7. Heather Widener | McLean, Virginia, USA |
| 8. Colin (Yowie) Bowey | Beechworth, Victoria, Australia |
| 9. Kelly Stubblefield | Mobile, Alabama, USA |