$(x + y) + (x^{2} + x y + y^{2}) + (x^{3} + x^{2} y + x y^{2} + y^{3}) + . . . n$ 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.
$\frac{x^{n} - y^{n}}{x - y} = x^{n - 1} + x^{n - 1} + x^{n - 2} y + . . . . + x y^{n - 2} + y^{n - 1}$
$L . H . S . = \frac{1}{x - y} [(x^{2} - y^{2}) + (x^{3} - y^{3}) + . . . . . n t e r m s] = \frac{1}{x - y} [S_{1} + S_{2} o f a G . P o f n t e r m s]$
$= \frac{1}{x - y} [S_{1} + S_{2}] o f a G . P o f n t e r m s = \frac{1}{x - y} [\frac{x^{2} (1 - x^{n})}{1 - x} - \frac{y^{2} (1 - y^{n})}{1 - y}]$
Note it may also be noted that $\frac{x^{3} + y^{3}}{x + y} = x^{2} - x y + y^{2} \frac{x^{5} + y^{5}}{x + y}$
$= x^{4} - x^{3} y + x^{2} y^{2} - x y^{3} + y^{4}$ and $\frac{x^{n} + y^{n}}{x + y} = x^{n - 1} - x^{n - 2} y + . . . - x y^{n - 2} + y^{n - 1}$ where n is 3,5,....i.e.odd.

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