Difference between revisions of "2001 AIME II Problems/Problem 3"
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<cmath>x_{531} + x_{753} + x_{975} = x_1 + x_3 + x_5 = x_1 + x_3 + (x_4 - x_3 + x_2 - x_1) = x_2 + x_4 = 375 + 523 = \boxed{898}.</cmath> | <cmath>x_{531} + x_{753} + x_{975} = x_1 + x_3 + x_5 = x_1 + x_3 + (x_4 - x_3 + x_2 - x_1) = x_2 + x_4 = 375 + 523 = \boxed{898}.</cmath> | ||
+ | |||
+ | Notice that we didn't need to use the values of <math>x_1</math> or <math>x_3</math> at all. | ||
== See also == | == See also == | ||
{{AIME box|year=2001|n=II|num-b=2|num-a=4}} | {{AIME box|year=2001|n=II|num-b=2|num-a=4}} | ||
{{MAA Notice}} | {{MAA Notice}} |
Revision as of 22:44, 30 December 2020
Contents
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.
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 |
The problems on this page are copyrighted by the Mathematical Association of America's American Mathematics Competitions.