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Difference between revisions of "2002 AIME II Problems/Problem 6"

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<math>250(-\frac{1}{9997} - \frac{1}{9998} - \frac{1}{9999} - \frac{1}{10000})</math> is not enough to bring <math>520.8</math> lower than <math>520.5</math>  so the answer is <math>\fbox{521}</math>
 
<math>250(-\frac{1}{9997} - \frac{1}{9998} - \frac{1}{9999} - \frac{1}{10000})</math> is not enough to bring <math>520.8</math> lower than <math>520.5</math>  so the answer is <math>\fbox{521}</math>
 +
 +
 +
If you didn't know <math>\frac{4}{n^2 - 4} = \frac{1}{n-2} - \frac{1}{n + 2}</math>, here's how you can find it out:
 +
 +
We know <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 thus: <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>
 +
 +
Then we have <math>\frac{1}{(n+2)(n-2)} = \frac{frac{1}{4}}{(n+2)} + \frac{=frac{-1}{4}}{(n+2)}</math> and we can continue as before.
  
 
== See also ==
 
== See also ==
 
{{AIME box|year=2002|n=II|num-b=5|num-a=7}}
 
{{AIME box|year=2002|n=II|num-b=5|num-a=7}}

Revision as of 16:12, 27 January 2009

Problem

Find the integer that is closest to $1000\sum_{n=3}^{10000}\frac1{n^2-4}$.

Solution

You know that $\frac{4}{n^2 - 4} = \frac{1}{n-2} - \frac{1}{n + 2}$.

So if you pull the $\frac{1}{4}$ out of the summation, you get

$250\sum_{n=3}^{10,000} (\frac{1}{n-2} - \frac{1}{n + 2})$.

Now that telescopes, leaving you with:

$250 (1 + \frac{1}{2} + \frac{1}{3} + \frac{1}{4} - \frac{1}{9997} - \frac{1}{9998} - \frac{1}{9999} - \frac{1}{10000}) = 250 + 125 + 83.3 + 62.5 - 250 (- \frac{1}{9997} - \frac{1}{9998} - \frac{1}{9999} - \frac{1}{10000})$

$250(-\frac{1}{9997} - \frac{1}{9998} - \frac{1}{9999} - \frac{1}{10000})$ is not enough to bring $520.8$ lower than $520.5$ so the answer is $\fbox{521}$


If you didn't know $\frac{4}{n^2 - 4} = \frac{1}{n-2} - \frac{1}{n + 2}$, here's how you can find it out:

We know $\frac{1}{n^2 - 4} = \frac{1}{(n+2)(n-2)}$. We can use the process of fractional decomposition to split this into two fractions thus: $\frac{1}{(n+2)(n-2)} = \frac{A}{(n+2)} + \frac{B}{(n+2)}$ for some A and B.

Solving for A and B gives $\1 = (n-2)A + (n+2)B$ (Error compiling LaTeX. Unknown error_msg) or $\1 = n(A+B)+ 2(B-A)$ (Error compiling LaTeX. Unknown error_msg). Since there is no n term on the left hand side, $\A+B=0$ (Error compiling LaTeX. Unknown error_msg) and by inspection $\1 = 2(B-A)$ (Error compiling LaTeX. Unknown error_msg). Solving yields $\A=frac{1}{4} B=frac{-1}{4}$ (Error compiling LaTeX. Unknown error_msg)

Then we have $\frac{1}{(n+2)(n-2)} = \frac{frac{1}{4}}{(n+2)} + \frac{=frac{-1}{4}}{(n+2)}$ and we can continue as before.

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
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