Difference between revisions of "2002 AIME I Problems/Problem 4"
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<math>\dfrac{1}{k^2+k}=\dfrac{1}{k(k+1)}=\dfrac{1}{k}-\dfrac{1}{k+1}</math>. Thus, | <math>\dfrac{1}{k^2+k}=\dfrac{1}{k(k+1)}=\dfrac{1}{k}-\dfrac{1}{k+1}</math>. Thus, | ||
− | <math>a_m+a_{m+1}++\cdots +a_{n-1}=\dfrac{1}{m}-\dfrac{1}{m+1}+\dfrac{1}{m+1}-\dfrac{1}{m+2}+\cdots +\dfrac{1}{n-1}-\dfrac{1}{n | + | <math>a_m+a_{m+1}++\cdots +a_{n-1}=\dfrac{1}{m}-\dfrac{1}{m+1}+\dfrac{1}{m+1}-\dfrac{1}{m+2}+\cdots +\dfrac{1}{n-1}-\dfrac{1}{n}=\dfrac{1}{m}-\dfrac{1}{n}</math> |
Which is | Which is | ||
− | <math>\dfrac{n | + | <math>\dfrac{m-n}{mn}=\dfrac{1}{29}</math> |
− | + | Since we need a 29 in the denominator, let <math>m=29t</math>*. Substituting, | |
− | <math> | + | <math>29t-n=nt</math> |
+ | <math>\frac{29t}{t+1} = n</math> | ||
− | + | Since n is an integer, <math>t+1 = 29</math>, or <math>t=28</math>. It quickly follows that <math>m=29(28)</math> and <math>n=28</math>, so <math>m+n = 30(28) = \fbox{840}</math>. | |
+ | |||
+ | *If <math>n=29t</math>, a similar argument to the one above implies <math>n=29(28)</math> and <math>m=28</math>, which implies <math>m>n</math>, which is impossible since <math>m-n>0</math>. | ||
== See also == | == See also == | ||
{{AIME box|year=2002|n=I|num-b=3|num-a=5}} | {{AIME box|year=2002|n=I|num-b=3|num-a=5}} |
Revision as of 16:13, 18 August 2008
Problem
Consider the sequence defined by for . Given that , for positive integers and with , find .
Solution
. Thus,
Which is
Since we need a 29 in the denominator, let *. Substituting,
Since n is an integer, , or . It quickly follows that and , so .
- If , a similar argument to the one above implies and , which implies , which is impossible since .
See also
2002 AIME I (Problems • Answer Key • Resources) | ||
Preceded by Problem 3 |
Followed by Problem 5 | |
1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 | ||
All AIME Problems and Solutions |