The first four terms in an arithmetic sequence are x + y, x - y, x + 3y, and 1, in that order. What is the fifth term?
Let the common difference be \(d\).
Because each term is the sum of the previous term and the common difference, we can write a system of equations:
\(x + y + d = x - y \ \ \ \ \ \ \ \ \ \ \ \ \ (i)\)
\(x - y + d = x + 3y \ \ \ \ \ \ \ \ \ \ \ (ii)\)
\(x + 3y + d = 1 \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ (iii)\)
From \((i)\), we have: \(y + d = -y\), meaning \(d = -2y\)
Plugging this into \((ii)\) and \((iii)\) gives us a system with 2 equations:
\(x - y - 2y = x + 3y\)
\(x + 3y + 2y = 1 \)
Solving the system, we find that \(x = 1\) and \(y = 0 \), meaning \(d = 0\), so the fifth term is \(1 + 0 = \color{brown}\boxed{1}\)