10.58 Let $S$ be the sum of the first nine terms of the sequence $x+a, x^{2}+2 a, x^{3}+3 a, \cdots$ $(x \neq 1)$, then $S$ equals
(A) $\frac{50 a+x+x^{8}}{x+1}$.
(B) $50 a-\frac{x+x^{10}}{x-1}$.
(C) $\frac{x^{9}-1}{x+1}+45 a$.
(D) $\frac{x^{10}-x}{x-1}+45 a$.
(E) $\frac{x^{11}-x}{x-1}+45 a$.
(10th American High School Mathematics Examination, 1959)
[Solution]
$$
\begin{aligned}
S & =(x+a)+\left(x^{2}+2 a\right)+\cdots+\left(x^{9}+9 a\right) \\
& =\left(x+x^{2}+\cdots+x^{9}\right)+(a+2 a+\cdots+9 a) \\
& =\frac{x\left(x^{9}-1\right)}{x-1}+\frac{9}{2}(a+9 a) \\
& =\frac{x^{10}-x}{x-1}+45 a .
\end{aligned}
$$

Therefore, the answer is $(D)$.