Difference between revisions of "2021 JMPSC Invitationals Problems/Problem 4"
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− | + | We notice that <math>x_5 + y_5 = 2x_4 + 2y_4 = 2(x_4 + y_4) = 2(2(x_3 + y_3)) = 2(2(2(x_2 + y_2))) = 2(2(2(2(x_1 + y_1)))) = 2(2(2(2(2(x_0 + y_0))))).</math> Given that <math>x_0 = 3</math> and <math>y_0 = 1</math> in the problem, we can plug this in to get that <math>x_5 + y_5 = 2(2(2(2(2(3 + 1))))) = 2(2(2(2(2(4))))) = 128.</math> | |
+ | |||
+ | We can use the same method to conclude that <math>x_5 - y_5 = 3(3(3(3(3(x_0 - y_0))))) = 3(3(3(3(3(3-1))))) = 3(3(3(3(3(2))))) = 486.</math> | ||
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+ | Adding this system of equations <math>x_5 + y_5 = 128</math> and <math>x_5 - y_5 = 486</math> gives us <math>2x_5 = 614.</math> Dividing both sides by <math>2</math>, results in <math>x_5 = 307.</math> | ||
+ | |||
+ | ~mahaler |
Revision as of 15:47, 11 July 2021
Problem
Let and
be sequences of real numbers such that
,
, and, for all positive integers
,
Find
.
Solution
We notice that Given that
and
in the problem, we can plug this in to get that
We can use the same method to conclude that
Adding this system of equations and
gives us
Dividing both sides by
, results in
~mahaler