Difference between revisions of "2005 AIME II Problems/Problem 7"
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== Problem == | == Problem == | ||
| − | Let <math> x=\frac{4}{(\sqrt{5}+1)(\sqrt[4]{5}+1)(\sqrt[8]{5}+1)(\sqrt[16]{5}+1)}. </math> Find <math> (x+1)^{48} | + | Let <math> x=\frac{4}{(\sqrt{5}+1)(\sqrt[4]{5}+1)(\sqrt[8]{5}+1)(\sqrt[16]{5}+1)}. </math> Find <math>\displaystyle (x+1)^{48}</math>. |
== Solution == | == Solution == | ||
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| − | It follows that <math>(x + 1)^{48} = (\sqrt[16]5)^{48} = 5^3 = 125</math> | + | It follows that <math>(x + 1)^{48} = (\sqrt[16]5)^{48} = 5^3 = 125</math>. |
== See Also == | == See Also == | ||
| − | + | {{AIME box|year=2005|n=II|num-b=6|num-a=8}} | |
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[[Category:Introductory Algebra Problems]] | [[Category:Introductory Algebra Problems]] | ||
Revision as of 18:55, 21 March 2007
Problem
Let ![$x=\frac{4}{(\sqrt{5}+1)(\sqrt[4]{5}+1)(\sqrt[8]{5}+1)(\sqrt[16]{5}+1)}.$](http://latex.artofproblemsolving.com/6/f/7/6f71d4755273a86295f916ae662a281e75af55c6.png)
Find 
.
Solution
We note that in general,
![$\displaystyle {} (\sqrt[2^n]{5} + 1)(\sqrt[2^n]{5} - 1) = (\sqrt[2^n]{5})^2 - 1^2 = \sqrt[2^{n-1}]{5} - 1$](http://latex.artofproblemsolving.com/e/f/c/efce902fd8efc20643c59a8fde2500a88952e03b.png)
.
It now becomes apparent that if we multiply the numerator and denominator of ![$\displaystyle \frac{4}{ (\sqrt{5}+1) (\sqrt[4]{5}+1) (\sqrt[8]{5}+1) (\sqrt[16]{5}+1) }$](http://latex.artofproblemsolving.com/3/1/a/31a02128c8c0f574d51feb9d4c4128dfca1dad75.png)
by ![$\displaystyle (\sqrt[16]{5} - 1)$](http://latex.artofproblemsolving.com/3/d/8/3d899f8802dd745f978572b19b4c3362c8e9eee8.png)
, the denominator will telescope to ![$\displaystyle \sqrt[1]{5} - 1 = 4$](http://latex.artofproblemsolving.com/b/2/2/b2240d179bebdcd1785041036e57dd80c688181f.png)
, so
![$\displaystyle x = \frac{4(\sqrt[16]{5} - 1)}{4} = \sqrt[16]{5} - 1$](http://latex.artofproblemsolving.com/1/7/7/17714c9e6e5feb3e6b059b13cbf2a25f57d359ac.png)
.
It follows that ![$(x + 1)^{48} = (\sqrt[16]5)^{48} = 5^3 = 125$](http://latex.artofproblemsolving.com/7/1/2/712816585976400e539a06ff0b568daa40f4e1da.png)
.
See Also
| 2005 AIME II (Problems • Answer Key • Resources) | ||
| Preceded by Problem 6 |
Followed by Problem 8 | |
| 1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 | ||
| All AIME Problems and Solutions | ||
