Art of Problem Solving
AoPS Online
Math texts, online classes, and more
for students in grades 5-12.
Visit AoPS Online ‚
Books for Grades 5-12 Online Courses
Beast Academy
Engaging math books and online learning
for students ages 6-13.
Visit Beast Academy ‚
Books for Ages 6-13 Beast Academy Online
AoPS Academy
Small live classes for advanced math
and language arts learners in grades 2-12.
Visit AoPS Academy ‚
Find a Physical Campus Visit the Virtual Campus

Difference between revisions of "2002 AIME I Problems/Problem 4"

(→‎Solution)
(→‎Solution)
Line 17: Line 17:
 
<math>\frac{29t}{t+1} = m</math>
 
<math>\frac{29t}{t+1} = m</math>
  
−
Since n is an integer, <math>t+1 = 29</math>, or <math>t=28</math>. It quickly follows that <math>n=29(28)</math> and <math>m=28</math>, so <math>m+n = 30(28) = \fbox{840}</math>.
+
Since m is an integer, <math>t+1 = 29</math>, or <math>t=28</math>. It quickly follows that <math>n=29(28)</math> and <math>m=28</math>, so <math>m+n = 30(28) = \fbox{840}</math>.
  
 
*If <math>m=29t</math>, a similar argument to the one above implies <math>m=29(28)</math> and <math>m=28</math>, which implies <math>m>n</math>, which is impossible since <math>n-m>0</math>.
 
*If <math>m=29t</math>, a similar argument to the one above implies <math>m=29(28)</math> and <math>m=28</math>, which implies <math>m>n</math>, which is impossible since <math>n-m>0</math>.

Revision as of 14:40, 5 March 2011

Problem

Consider the sequence defined by $a_k =\dfrac{1}{k^2+k}$ for $k\geq 1$. Given that $a_m+a_{m+1}+\cdots+a_{n-1}=\dfrac{1}{29}$, for positive integers $m$ and $n$ with $m<n$, find $m+n$.

Solution

$\dfrac{1}{k^2+k}=\dfrac{1}{k(k+1)}=\dfrac{1}{k}-\dfrac{1}{k+1}$. Thus,

$a_m+a_{m+1}++\cdots +a_{n-1}=\dfrac{1}{m}-\dfrac{1}{m+1}+\dfrac{1}{m+1}-\dfrac{1}{m+2}+\cdots +\dfrac{1}{n-1}-\dfrac{1}{n}=\dfrac{1}{m}-\dfrac{1}{n}$

Which is

$\dfrac{n-m}{mn}=\dfrac{1}{29}$

Since we need a 29 in the denominator, let $n=29t$.* Substituting,

$29t-m=mt$

$\frac{29t}{t+1} = m$

Since m is an integer, $t+1 = 29$, or $t=28$. It quickly follows that $n=29(28)$ and $m=28$, so $m+n = 30(28) = \fbox{840}$.

  • If $m=29t$, a similar argument to the one above implies $m=29(28)$ and $m=28$, which implies $m>n$, which is impossible since $n-m>0$.

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
Retrieved from "https://artofproblemsolving.com/wiki/index.php?title=2002_AIME_I_Problems/Problem_4&oldid=37245"