Difference between revisions of "2003 AMC 10B Problems/Problem 24"

(Created page with "==Problem== The first four terms in an arithmetic sequence are <math>x+y</math>,<math>x-y</math> ,<math>xy</math> , and <math>\frac{x}{y}</math>, in that order. What is the fi...")
 
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==Solution==
 
==Solution==
  
We get the equations:
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The difference between consecutive terms is <math>(x-y)-(x+y)=-2y.</math> Therefore we can also express the third and fourth terms as <math>x-3y</math> and <math>x-5y.</math> Then we can set them equal to <math>xy</math> and <math>\frac{x}{y}</math> because they are the same thing.
  
<cmath> \begin{align*}x+y+d&=x-y\\ x-y+d&=xy\\ xy+d &=\frac{x}{y}\end{align*} </cmath>
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<cmath>\begin{align*}
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xy&=x-3y\\
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xy-x&=-3y\\
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x(y-1)&=-3y\\
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x&=\frac{-3y}{y-1}
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\end{align*}</cmath>
  
Simplifying and rearranging, we can keep substituting and find that <math> (d, x, y)=\left(\frac{6}{5},-\frac{9}{8},-\frac{3}{5}\right) </math>
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Substitute into our other equation.
  
Finally, <cmath> \frac{x}{y}+d=\frac{9\cdot 5}{8\cdot 3}+\frac{6}{5}=\frac{123}{40}\rightarrow\boxed{\textbf{E}}. </cmath>
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<cmath>\begin{align*}
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\frac{x}{y}&=x-5y\\
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\frac{-3}{y-1}&=\frac{-3y}{y-1}-5y\\
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-3&=-3y-5y(y-1)\\
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0&=5y^2-2y-2\\
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0&=(5y+3)(y-1)\\
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y&=-\frac35, 1</cmath>
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 +
But <math>y</math> cannot be <math>1</math> because then every term would be equal to <math>x.</math> Therefore <math>y=-\frac35.</math> Substituting the value for <math>y</math> into any of the equations, we get <math>x=-\frac98.</math> Finally,
 +
 
 +
<cmath> \frac{x}{y}-2y=\frac{9\cdot 5}{8\cdot 3}+\frac{6}{5}=\boxed{\textbf{(E)}\ \frac{123}{40}}</cmath>
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==See Also==
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{{AMC10 box|year=2003|ab=B|num-b=23|num-a=25}}

Revision as of 22:15, 26 November 2011

Problem

The first four terms in an arithmetic sequence are $x+y$,$x-y$ ,$xy$ , and $\frac{x}{y}$, in that order. What is the fifth term?

$\textbf{(A)}\ -\frac{15}{8}\qquad\textbf{(B)}\ -\frac{6}{5}\qquad\textbf{(C)}\ 0\qquad\textbf{(D)}\ \frac{27}{20}\qquad\textbf{(E)}\ \frac{123}{40}$


Solution

The difference between consecutive terms is $(x-y)-(x+y)=-2y.$ Therefore we can also express the third and fourth terms as $x-3y$ and $x-5y.$ Then we can set them equal to $xy$ and $\frac{x}{y}$ because they are the same thing.

\begin{align*} xy&=x-3y\\ xy-x&=-3y\\ x(y-1)&=-3y\\ x&=\frac{-3y}{y-1} \end{align*}

Substitute into our other equation.

\begin{align*}
\frac{x}{y}&=x-5y\\
\frac{-3}{y-1}&=\frac{-3y}{y-1}-5y\\
-3&=-3y-5y(y-1)\\
0&=5y^2-2y-2\\
0&=(5y+3)(y-1)\\
y&=-\frac35, 1 (Error compiling LaTeX. Unknown error_msg)

But $y$ cannot be $1$ because then every term would be equal to $x.$ Therefore $y=-\frac35.$ Substituting the value for $y$ into any of the equations, we get $x=-\frac98.$ Finally,

\[\frac{x}{y}-2y=\frac{9\cdot 5}{8\cdot 3}+\frac{6}{5}=\boxed{\textbf{(E)}\ \frac{123}{40}}\]

See Also

2003 AMC 10B (ProblemsAnswer KeyResources)
Preceded by
Problem 23
Followed by
Problem 25
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25
All AMC 10 Problems and Solutions