Sum the series : $x (x + y) + x^{2} (x^{2} + y^{2}) + x^{3} (x^{3} + y^{3}) + . .$ to n terms

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Solution

$x (x + y) + x^{2} (x^{2} + y^{2}) + x^{3} (x^{3} + y^{3}) + . . . . . + x^{n} (x^{n} + y^{n})$

$= (x^{2} + x^{4} + x^{6} + . . . . . + x^{2 n}) + (x y + x^{2} y^{2} + x^{3} y^{3} + . . . . . + x^{n} y^{n})$

$= \frac{x^{2} (x^{2 n} - 1)}{x^{2} - 1} + \frac{x y (x^{n} y^{n} - 1)}{x y - 1}$

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