2001 AIME II Problems/Problem 3
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Contents
[hide]Problem
Given that
find the value of .
Solution
We find that by the recursive formula. Summing the recursions
yields . Thus . Since , it follows that
Solution Variant
The recursive formula suggests telescoping. Indeed, if we add and , we have .
Subtracting yields .
Thus,
Notice that we didn't need to use the values of or at all.
Non-Rigorous Solution
Calculate the first few terms:
At this point it is pretty clear that the sequence is periodic with period 10 (one may prove it quite easily like in solution 1) so our answer is obviously
Video Solution by OmegaLearn
https://youtu.be/lH-0ul1hwKw?t=870
~ pi_is_3.14
See also
2001 AIME II (Problems • Answer Key • Resources) | ||
Preceded by Problem 2 |
Followed by Problem 4 | |
1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 | ||
All AIME Problems and Solutions |
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