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# Arithmetic sequence

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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?

Jul 26, 2022

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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 \(1 + 0 = \color{brown}\boxed{1}\)

Jul 26, 2022