Given the following table of equations, write out the line that begins with 3773 using the same pattern.
13 + 23 - 13 = 8
23 + 33 - 13 = 34
33 + 53 - 23 = 144
53 + 83 - 33 = 610
83 + 133 - 53 = 2584
:
:
:
3773 + _____ - _____ = _____
What is the significance of these numbers?
Solution to the Problem:
3773 + 6103 - 2333 = 267914296These are Fibonacci numbers, and the pattern is:
(2nd Fibonacci #)3 + (3rd Fibonacci #)3 - (1st Fibonacci #)3 = 6th Fibonacci #
(3rd Fibonacci #)3 + (4th Fibonacci #)3 - (2nd Fibonacci #)3 = 9th Fibonacci #
(4th Fibonacci #)3 + (5th Fibonacci #)3 - (3rd Fibonacci #)3 = 12th Fibonacci #
(5th Fibonacci #)3 + (6th Fibonacci #)3 - (4th Fibonacci #)3 = 15th Fibonacci #
(6th Fibonacci #)3 + (7th Fibonacci #)3 - (5th Fibonacci #)3 = 18th Fibonacci #
:
:
:
(14th Fibonacci #)3 + (15th Fibonacci #)3 - (13th Fibonacci #)3 = 42nd Fibonacci #
Did you notice that in the last line 14 + 15 + 13 = 42
and that it works for every line?
In general,
(n+1st Fibonacci #)3 + (n+2nd Fibonacci #)3 - (nth Fibonacci #)3 =
(n + (n+1) + (n+2)) Fibonacci #
Correctly solved by:
| 1. Kate Cotter | Elmhurst, Illinois |
| 2. Mike Bova | Wallingford Connecticut |
| 3. James Alarie | Flint, Michigan |
| 4. krishnanrk2000@yahoo.com | ---------- |
| 5. Keith Mealy | Cincinnati, Ohio |
| 6. Elisa Arostegui | Carmichael, California |
| 7. Chad Fore | Gate City, Virginia |