Difference between revisions of "Mock AIME 2 2006-2007 Problems/Problem 5"
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− | Given that <math> | + | ==Problem== |
+ | Given that <math> iz^2=1+\frac 2z + \frac{3}{z^2}+\frac{4}{z ^3}+\frac{5}{z^4}+\cdots</math> and <math>z=n\pm \sqrt{-i},</math> find <math> \lfloor 100n \rfloor</math>. | ||
+ | |||
+ | ==Solution== | ||
+ | Multiplying both sides of the equation by <math>z</math>, we get | ||
+ | <center><math>iz^3 = z + 2 + \frac{3}{z} + \frac{4}{z^2} + \cdots</math>,</center> | ||
+ | and subtracting the original equation from this one we get | ||
+ | <center><math>iz^2(z-1)=z+1+\frac{1}{z}+\frac{1}{z^2}+\frac{1}{z^3}+\cdots</math>.</center> | ||
+ | Using the formula for an infinite geometric series, we find | ||
+ | <center><math>iz^2(z-1)=\frac{z}{1-\frac{1}{z}}=\frac{z^2}{z-1}</math>.</center> | ||
+ | Rearranging, we get | ||
+ | <center><math>iz^2(z-1)^2=z^2\iff (z-1)^2=\frac{1}{i}=-i\Rightarrow z=1\pm\sqrt{-i}</math>.</center> | ||
+ | Thus <math>n=1</math>, and the answer is <math>\lfloor 100n \rfloor = \boxed{100}</math>. | ||
+ | |||
+ | ==See also== | ||
+ | {{Mock AIME box|year=2006-2007|n=2|num-b=4|num-a=6}} |
Latest revision as of 10:51, 4 April 2012
Problem
Given that and
find
.
Solution
Multiplying both sides of the equation by , we get
![$iz^3 = z + 2 + \frac{3}{z} + \frac{4}{z^2} + \cdots$](http://latex.artofproblemsolving.com/7/3/d/73dd020de5a5408e6f52c937e7fa8d3ad55908ad.png)
and subtracting the original equation from this one we get
![$iz^2(z-1)=z+1+\frac{1}{z}+\frac{1}{z^2}+\frac{1}{z^3}+\cdots$](http://latex.artofproblemsolving.com/4/8/0/480103d40eb8a07b9f7e95e6b28f7e18bebacdf5.png)
Using the formula for an infinite geometric series, we find
![$iz^2(z-1)=\frac{z}{1-\frac{1}{z}}=\frac{z^2}{z-1}$](http://latex.artofproblemsolving.com/f/c/d/fcd64829e67ebcc4eab6aa004f4ebf575367775d.png)
Rearranging, we get
![$iz^2(z-1)^2=z^2\iff (z-1)^2=\frac{1}{i}=-i\Rightarrow z=1\pm\sqrt{-i}$](http://latex.artofproblemsolving.com/c/9/9/c990fcfac5ea595298da9e3f9b72af3d76bdd42f.png)
Thus , and the answer is
.
See also
Mock AIME 2 2006-2007 (Problems, Source) | ||
Preceded by Problem 4 |
Followed by Problem 6 | |
1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 |