Difference between revisions of "2001 AIME II Problems/Problem 3"
I like pie (talk | contribs) (eqnarray -> align; LaTeX style) |
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== Problem == | == Problem == | ||
Given that | Given that | ||
| − | < | + | <cmath> |
| + | \begin{align*}x_{1}&=211,\\ | ||
| + | x_{2}&=375,\\ | ||
| + | x_{3}&=420,\\ | ||
| + | x_{4}&=523,\ \text{and}\\ | ||
| + | x_{n}&=x_{n-1}-x_{n-2}+x_{n-3}-x_{n-4}\ \text{when}\ n\geq5, \end{align*} | ||
| + | </cmath> | ||
find the value of <math>x_{531}+x_{753}+x_{975}</math>. | find the value of <math>x_{531}+x_{753}+x_{975}</math>. | ||
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Therefore, <math>x_{y}=x_{y-10}</math>, so | Therefore, <math>x_{y}=x_{y-10}</math>, so | ||
| − | + | <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_4+x_2=523+420=\boxed{943}</cmath> | + | \begin{align*}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_4+x_2=523+420=\boxed{943}\end{align*} | ||
| + | </cmath> | ||
== 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}} | ||
find the value of 
.
, so
