Difference between revisions of "2002 AIME II Problems/Problem 6"
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== Solution 1 == | == Solution 1 == | ||
− | We know that <math>\frac{ | + | We know that <math>\frac{1}{n^2 - 4} = \frac{1}{(n+2)(n-2)}</math>. We can use the process of fractional decomposition to split this into two fractions: <math>\frac{1}{(n+2)(n-2)} = \frac{A}{(n+2)} + \frac{B}{(n-2)}</math> for some A and B. |
− | + | Solving for A and B gives <math>1 = (n-2)A + (n+2)B</math> or <math>1 = n(A+B)+ 2(B-A)</math>. Since there is no n term on the left hand side, <math> A+B=0</math> and by inspection <math>1 = 2(B-A)</math>. Solving yields <math> A=\frac{1}{4}, B=\frac{-1}{4}</math> | |
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+ | Therefore, <math>\frac{1}{n^2-4} = \frac{1}{(n+2)(n-2)} = \frac{ \frac{1}{4} }{(n-2)} + \frac{ \frac{-1}{4} }{(n+2)} = \frac{1}{4} \left( \frac{1}{n-2} - \frac{1}{n+2} \right)</math>. | ||
− | <math>250\sum_{n=3}^{10,000} (\frac{1}{n-2} - \frac{1}{n + 2})</math>. | + | And so, <math>1000\sum_{n=3}^{10,000} \frac{1}{n^2-4} = 1000\sum_{n=3}^{10,000} \frac{1}{4} \left( \frac{1}{n-2} - \frac{1}{n+2} \right) = 250\sum_{n=3}^{10,000} (\frac{1}{n-2} - \frac{1}{n + 2})</math>. |
− | + | This telescopes into: | |
− | <math>250 (1 + \frac{1}{2} + \frac{1}{3} + \frac{1}{4} - \frac{1}{9999} - \frac{1}{10000} - \frac{1}{10001} - \frac{1}{10002}) = 250 + 125 + 83.3 + 62.5 - 250 ( | + | <math>250 (1 + \frac{1}{2} + \frac{1}{3} + \frac{1}{4} - \frac{1}{9999} - \frac{1}{10000} - \frac{1}{10001} - \frac{1}{10002}) = 250 + 125 + 83.3 + 62.5 - 250 (\frac{1}{9999} + \frac{1}{10000} + \frac{1}{10001} + \frac{1}{10002})</math> |
The small fractional terms are not enough to bring <math>520.8</math> lower than <math>520.5,</math> so the answer is <math>\fbox{521}</math> | The small fractional terms are not enough to bring <math>520.8</math> lower than <math>520.5,</math> so the answer is <math>\fbox{521}</math> | ||
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== Solution 2 == | == Solution 2 == | ||
− | Using the fact that <math>\frac{1}{n(n+k)} = \frac{1}{k} ( \frac{1}{n}-\frac{1}{n+k} )</math> or by partial fraction decomposition, we both obtained <math>\frac{1}{x^2-4} = \frac{1}{4}(\frac{1}{x-2}-\frac{1}{x+2})</math>. The denominators of the positive terms are <math>1,2,..,9998</math>, while the negative ones are <math>5,6,...,10002</math>. Hence we are left with <math>1000 \cdot 4 (1 + \frac{1}{2} + \frac{1}{3} + \frac{1}{4} - \frac{1}{9999} - \frac{1}{10000} - \frac{1}{10001} - \frac{1}{10002})</math>. We can simply ignore the last <math>4</math> terms, and we get it is approximately <math>1000*\frac{25}{48}</math>. Computing <math>\frac{25}{48}</math> which is about <math>0.5208..</math> and moving the decimal point three times, we get that the answer is <math>521</math> | + | Using the fact that <math>\frac{1}{n(n+k)} = \frac{1}{k} ( \frac{1}{n}-\frac{1}{n+k} )</math> or by partial fraction decomposition, we both obtained <math>\frac{1}{x^2-4} = \frac{1}{4}(\frac{1}{x-2}-\frac{1}{x+2})</math>. The denominators of the positive terms are <math>1,2,..,9998</math>, while the negative ones are <math>5,6,...,10002</math>. Hence we are left with <math>1000 \cdot \frac{1}{4} (1 + \frac{1}{2} + \frac{1}{3} + \frac{1}{4} - \frac{1}{9999} - \frac{1}{10000} - \frac{1}{10001} - \frac{1}{10002})</math>. We can simply ignore the last <math>4</math> terms, and we get it is approximately <math>1000*\frac{25}{48}</math>. Computing <math>\frac{25}{48}</math> which is about <math>0.5208..</math> and moving the decimal point three times, we get that the answer is <math>521</math> |
== See also == | == See also == |
Latest revision as of 05:56, 23 August 2022
Contents
Problem
Find the integer that is closest to .
Solution 1
We know that . We can use the process of fractional decomposition to split this into two fractions:
for some A and B.
Solving for A and B gives or
. Since there is no n term on the left hand side,
and by inspection
. Solving yields
Therefore, .
And so, .
This telescopes into:
The small fractional terms are not enough to bring lower than
so the answer is
Solution 2
Using the fact that or by partial fraction decomposition, we both obtained
. The denominators of the positive terms are
, while the negative ones are
. Hence we are left with
. We can simply ignore the last
terms, and we get it is approximately
. Computing
which is about
and moving the decimal point three times, we get that the answer is
See also
2002 AIME II (Problems • Answer Key • Resources) | ||
Preceded by Problem 5 |
Followed by Problem 7 | |
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.