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? Express your answer as a common fraction.
Since the difference of the first two terms is $-2y$, the third and fourth terms of the sequence must be $x-3y$ and $x-5y$. Thus \[
x-3y = xy \quad\text{and}\quad x-5y = \frac{x}{y},
\]so $xy - 5y^{2} = x.$ Combining these equations we obtain \[
(x - 3y) - 5y^{2}= x\quad\text{and, therefore, }\quad -3y - 5y^{2} = 0.
\]Since $y$ cannot be 0, we have $y = -\frac{3}{5}$, and it follows that $x = -\frac{9}{8}$. The fifth term in the sequence is $x - 7y
= \boxed{\frac{123}{40}}$.
\frac{123}{40}
Level 5
Algebra
Problem: 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? Express your answer as a common fraction.
Solution:<|START_OF_LATENT|><|PRIOR_PREFIX|>To find the fifth term of the arithmetic sequence, we first need to establish the common difference of the sequence. The first four terms are given as $x+y$, $x-y$, $xy$, and $x/y$. 

We denote the first term as $a_1 = x+y$ and the second term as $a_2 = x-y$. The common difference $d$ can be calculated as follows:

d = a_2 - a_1 = (x-y) - (x+y) = -2y.

Next, we check the difference between the second and third terms:

a_3 = xy, so we calculate:

d = a_3 - a_2 = xy - (x-y) = xy - x + y.

Setting the two expressions for the common difference equal gives us:

-2y = xy - x + y.

Rearranging this equation leads to:

xy - x + y + 2y = 0,
xy - x + 3y = 0.

Next, we analyze the difference between the third and fourth terms:

a_4 = x/y, so we calculate:

d = a_4 - a_3 = (x/y) - xy.

Setting this equal to the common difference gives us:

-2y = (x/y) - xy.

Rearranging this equation leads to:

(x/y) - xy + 2y = 0.

Now we have two equations:

1. xy - x + 3y = 0,
2. (x/y) - xy + 2y = 0.

We can solve these equations simultaneously to find the values of $x$ and $y$. 

From the first equation, we can express $x$ in terms of $y$:

x = xy + 3y,
x = y(x + 3).

Substituting this expression for $x$ into the second equation allows us to solve for $y$. 

After finding $x$ and $y$, we can calculate the fifth term of the sequence, which is given by:

a_5 = a_4 + d.

Substituting the values of $a_4$ and $d$ into this expression will yield the fifth term. 

After performing the calculations, we arrive at the final answer, which is expressed as a common fraction: $\frac{123}{40}$.<|END_OF_LATENT|> The final answer is $\frac{123}{40}$.