Difference between revisions of "Mock AIME 2 Pre 2005 Problems/Problem 2"
(Created page with "== Problem == <math>x</math> is a real number with the property that <math>x+\tfrac1x = 3</math>. Let <math>S_m = x^m + \tfrac{1}{x^m}</math>. Determine the value of <math>S_7...") |
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== Solution == | == Solution == | ||
We can calculate <cmath>x^2 + \dfrac{1}{x^2} = \left(x + \dfrac{1}{x}\right)^2 - 2 = 3^2 -2 = 7.</cmath> Similarly, <cmath>x^3 + \dfrac{1}{x^3} = \left(x + \dfrac{1}{x}\right) \left(x^2 + \dfrac{1}{x^2}\right) - \left(x + \dfrac{1}{x}\right) = 3 \cdot 7 - 3 = 18</cmath> and <cmath>x^4 + \dfrac{1}{x^4} = \left(x^2 + \dfrac{1}{x^2}\right)^2 - 2 = 7^2 - 2 = 47.</cmath> Finally, <cmath>x^7 + \dfrac{1}{x^7} = \left(x^3 + \dfrac{1}{x^3}\right) \left(x^4 + \dfrac{1}{x^4}\right) - \left(x + \dfrac{1}{x}\right) = 18 \cdot 47 - 3 = \boxed{843}.</cmath> | We can calculate <cmath>x^2 + \dfrac{1}{x^2} = \left(x + \dfrac{1}{x}\right)^2 - 2 = 3^2 -2 = 7.</cmath> Similarly, <cmath>x^3 + \dfrac{1}{x^3} = \left(x + \dfrac{1}{x}\right) \left(x^2 + \dfrac{1}{x^2}\right) - \left(x + \dfrac{1}{x}\right) = 3 \cdot 7 - 3 = 18</cmath> and <cmath>x^4 + \dfrac{1}{x^4} = \left(x^2 + \dfrac{1}{x^2}\right)^2 - 2 = 7^2 - 2 = 47.</cmath> Finally, <cmath>x^7 + \dfrac{1}{x^7} = \left(x^3 + \dfrac{1}{x^3}\right) \left(x^4 + \dfrac{1}{x^4}\right) - \left(x + \dfrac{1}{x}\right) = 18 \cdot 47 - 3 = \boxed{843}.</cmath> | ||
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== See also == | == See also == | ||
is a real number with the property that 
. Let 
. Determine the value of 
.
![\[x^2 + \dfrac{1}{x^2} = \left(x + \dfrac{1}{x}\right)^2 - 2 = 3^2 -2 = 7.\]](http://latex.artofproblemsolving.com/b/b/5/bb53b97e3e0f441767043012f0e72a5001594759.png)
Similarly, ![\[x^3 + \dfrac{1}{x^3} = \left(x + \dfrac{1}{x}\right) \left(x^2 + \dfrac{1}{x^2}\right) - \left(x + \dfrac{1}{x}\right) = 3 \cdot 7 - 3 = 18\]](http://latex.artofproblemsolving.com/8/1/d/81d52885045a282be85eb4ad0e116ecf0dc50eb9.png)
and ![\[x^4 + \dfrac{1}{x^4} = \left(x^2 + \dfrac{1}{x^2}\right)^2 - 2 = 7^2 - 2 = 47.\]](http://latex.artofproblemsolving.com/4/a/5/4a5d8cb8f4559bcb7f69faf8971ee712364ced6b.png)
Finally, ![\[x^7 + \dfrac{1}{x^7} = \left(x^3 + \dfrac{1}{x^3}\right) \left(x^4 + \dfrac{1}{x^4}\right) - \left(x + \dfrac{1}{x}\right) = 18 \cdot 47 - 3 = \boxed{843}.\]](http://latex.artofproblemsolving.com/d/d/b/ddba4f22258b80914f836b278a6901de20c3685f.png)
