Difference between revisions of "Mock AIME 1 Pre 2005 Problems/Problem 13"
Rdj5933mile5 (talk | contribs) (Created page with '=Solution= First we consider the fact that <math> 7R_{n} +2R_{n-1}-9R_{n-2}= 64 </math> Now, consider the fact that <math>S= \frac{R_0}{1} +\frac{R_1}{2} +\frac{R_2}{4} + \f…') |
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| − | = | + | ==Problem== |
| + | A sequence <math> \{R_{n}\}_{n\ge 0} </math> obeys the recurrence <math> 7R_{n}= 64-2R_{n-1}+9R_{n-2} </math> for any integers <math>n\ge 2</math>. Additionally, <math>R_0=10</math> and <math>R_1=-2</math>. Let <cmath> S =\sum_{i=0}^{\infty}\frac{R_{i}}{2^{i}} </cmath> | ||
| + | <math>S</math> can be expressed as <math>\frac{m}{n}</math> for two 2 relatively prime numbers <math>m</math> and <math>n</math>. Determine the value of <math>m+n</math> | ||
| − | First we consider the fact that | + | ==Solution== |
| + | |||
| + | First, we consider the fact that | ||
<math> 7R_{n} +2R_{n-1}-9R_{n-2}= 64 </math> | <math> 7R_{n} +2R_{n-1}-9R_{n-2}= 64 </math> | ||
Latest revision as of 13:22, 26 March 2011
Problem
A sequence 
obeys the recurrence 
for any integers 
. Additionally, 
and 
. Let ![\[S =\sum_{i=0}^{\infty}\frac{R_{i}}{2^{i}}\]](http://latex.artofproblemsolving.com/d/1/9/d19b0ce756324b08284c4c9084c64764baf2fa55.png)
can be expressed as 
for two 2 relatively prime numbers 
and 
. Determine the value of 
Solution
First, we consider the fact that
Now, consider the fact that
Thus,
and
and
Adding these together we get that
Since 420 and 23 are relatively prime we find that the answer is 
