2003 AMC 10B Problem 24

Attempt Problem 24 of the 2003 AMC 10B below, then check your answer against the professionally curated solution from LIVE by Po-Shen Loh. You can also try the full timed exam, view all 2003 AMC 10B solutions, or check the answer key.

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24.

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

$-\dfrac{15}{8}$

$-\dfrac{6}{5}$

$0$

$\dfrac{27}{20}$

$\dfrac{123}{40}$

The common difference is $-2y,$ so the third and fourth terms must be $x-3y$ and $x-5y.$ Thus $xy=x-3y$ and $\dfrac{x}{y}=x-5y.$

From $\dfrac{x}{y}=x-5y$ we get $x=xy-5y^2,$ and substituting $xy=x-3y$ gives $-3y-5y^2=0.$ Since $y\ne 0,$ $y=-\dfrac35$ and then $x=-\dfrac98.$

The fifth term is $x-7y=-\dfrac98 + \dfrac{21}{5}=\dfrac{123}{40}.$

Thus, the correct answer is E.