Difference between revisions of "2006 AIME A Problems/Problem 15"
5849206328x (talk | contribs) m (→Solution: Fixed typo) |
5849206328x (talk | contribs) |
||
| Line 1: | Line 1: | ||
== Problem == | == Problem == | ||
Given that a sequence satisfies <math> x_0=0 </math> and <math> |x_k|=|x_{k-1}+3| </math> for all integers <math> k\ge 1, </math> find the minimum possible value of <math> |x_1+x_2+\cdots+x_{2006}|. </math> | Given that a sequence satisfies <math> x_0=0 </math> and <math> |x_k|=|x_{k-1}+3| </math> for all integers <math> k\ge 1, </math> find the minimum possible value of <math> |x_1+x_2+\cdots+x_{2006}|. </math> | ||
| + | |||
| + | {{Delete|The correct page that is linked to contains this solution and others}} | ||
== Solution == | == Solution == | ||
Revision as of 12:53, 23 May 2009
Problem
Given that a sequence satisfies 
and 
for all integers 
find the minimum possible value of 
| This article has been proposed for deletion. The reason given is: The correct page that is linked to contains this solution and others.
Sysops: Before deleting this article, please check the article discussion pages and history. |
Solution
If you take the recursion formula and square both sides, you get 
, or 
. If you take the sum of this from 1 to 2007, you see that
x_{2007}^2-x_0^2=6(x_0+x_1+x_2+\cdots +x_{2006})+9*2007\\ x_{2007}^2=6(S)+9*2007\\
S=\frac{x_{2007}^2-9*2007}{6}\end{eqnarray*}$ (Error compiling LaTeX. Unknown error_msg)where 
.
Note that 
is a multiple of 3, and the closest square multiple of 3 to 2*2007 is 
. Therefore the smallest value of the absolute value of 
is
.
See also
| 2006 AIME II (Problems • Answer Key • Resources) | ||
| Preceded by Problem 14 |
Followed by Final Problem | |
| 1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 | ||
| All AIME Problems and Solutions | ||
