1996 AJHSME Problems/Problem 20
Problem
Suppose there is a special key on a calculator that replaces the number currently displayed with the number given by the formula . For example, if the calculator is displaying 2 and the special key is pressed, then the calculator will display -1 since . Now suppose that the calculator is displaying 5. After the special key is pressed 100 times in a row, the calculator will display
Solution
We look for a pattern, hoping this sequence either settles down to one number, or that it forms a cycle that repeats.
After press, the calculator displays
After presses, the calculator displays
After presses, the calculator displays
Thus, every three presses, the display will be . On press , the display will be . One more press will give , which is answer .
See Also
1996 AJHSME (Problems • Answer Key • Resources) | ||
Preceded by Problem 19 |
Followed by Problem 21 | |
1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 • 16 • 17 • 18 • 19 • 20 • 21 • 22 • 23 • 24 • 25 | ||
All AJHSME/AMC 8 Problems and Solutions |
The problems on this page are copyrighted by the Mathematical Association of America's American Mathematics Competitions.