Find $(x + y) + (x^{2} + x y + y^{2}) + (x^{3} + x^{2} y + {x y}^{2} + y^{3}) + . . .$ to $n$ terms,

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Solution

It is easy to observe that
$\frac{x^{2} - y^{2}}{x - y} = x + y$ , $\frac{x^{3} - y^{3}}{x - y} = x^{2} + x y + y^{2}$ , etc.
$\frac{x^{n} - y^{n}}{x - y} = x^{n - 1} + x^{n - 2} y + . . . . + x y^{n - 2} + y^{n - 1}$
$L . H . S . = \frac{1}{x - y} [(x^{2} - y^{2}) + (x^{3} - y^{3}) + . . . . n t e r m s]$
$= \frac{1}{x - y} (S_{1} - S_{2})$ of a $G . P .$ of $n$ terms
$= \frac{1}{x - y} [\frac{x^{2} (1 - x^{n})}{1 - x} - \frac{y^{2} (1 - y^{n})}{1 - y}]$

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