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Question
Find the sum of the following arithmetic progression :
\[\frac{x - y}{x + y}, \frac{3x - 2y}{x + y}, \frac{5x - 3y}{x + y}\], ... to n terms.
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Solution
\[\frac{x - y}{x + y}, \frac{3x - 2y}{x + y}, \frac{5x - 3y}{x + y}\] ... to n terms
\[\text { We have:} \]
\[ a = \frac{x - y}{x + y}, d = $\left( \frac{3x - 2y}{x + y} - \frac{x - y}{x + y} \right)$ = \left( \frac{2x - y}{x + y} \right)\]
\[ S_n = \frac{n}{2}\left[ 2a + (n - 1)d \right]\]
\[ = \frac{n}{2}\left[ 2\left( \frac{x - y}{x + y} \right) + (n - 1)\left( \frac{2x - y}{x + y} \right) \right]\]
\[ = \frac{n}{2(x + y)}\left[ (2x - 2y) + (2x - y)(n - 1) \right]\]
\[ = \frac{n}{2(x + y)}\left[ 2x - 2y - 2x + y + n(2x - y) \right]\]
\[ = \frac{n}{2(x + y)}\left[ n(2x - y) - y \right]\]
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