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Synthetic Division of Polynomials

Evaluate : ...

Question

Evaluate : $\frac{x + x^{2} + x^{3} + . . . . . . + x^{10}}{x - 1}$

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Solution

$x, x^{2}, x^{3}, . . ., x^{10}$ are in G.P
where $a = x, r = \frac{x^{2}}{x} = x$
$∴ S_{n} = \frac{a (r^{n} - 1)}{r - 1}$
$S = \frac{x + x^{2} + . . . . x^{10}}{x - 1} = \frac{x (x^{10} - 1)}{\frac{x - 1}{x - 1}} = \frac{x (x^{10} - 1)}{(x - 1)^{2}}$

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