1990 AIME Problems/Problem 15
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Contents
[hide]Problem
Find if the real numbers
,
,
, and
satisfy the equations
Solution 1
Set and
. Then the relationship
can be exploited:
Therefore:
Consequently, and
. Finally:
Solution 2
A recurrence of the form will have the closed form
, where
are the values of the starting term that make the sequence geometric, and
are the appropriately chosen constants such that those special starting terms linearly combine to form the actual starting terms.
Suppose we have such a recurrence with and
. Then
, and
.
Solving these simultaneous equations for and
, we see that
and
. So,
.
See also
1990 AIME (Problems • Answer Key • Resources) | ||
Preceded by Problem 14 |
Followed by Last question | |
1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 | ||
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