1998 AIME Problems/Problem 8
Problem
Except for the first two terms, each term of the sequence is obtained by subtracting the preceding term from the one before that. The last term of the sequence is the first negative term encounted. What positive integer produces a sequence of maximum length?
Solution
The best way to start is to just write out some terms.
0 | 1 | 2 | 3 | 4 | 5 | 6 |
aa | aaa | a |
It is now apparent that each term can be written as
where the are Fibonacci numbers. This can be proven through induction.
Solution 1
We can start to write out some of the inequalities now:
And in general,
It is apparent that the bounds are slowly closing in on , so we can just calculate for some large value of (randomly, 10, 11):
Thus the sequence is maximized when
Solution 2
It is well known that , so approaches
See also
1998 AIME (Problems • Answer Key • Resources) | ||
Preceded by Problem 7 |
Followed by Problem 9 | |
1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 | ||
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