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smc

algebra senior

Problem

Let S be the sum of the first nine terms of the sequence

x + a, x^{2} + 2 a, x^{3} + 3 a, \dots .

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

Solution

We know

x + x^{2} \dots + x^{9} = x (1 + x \dots + x^{8}) = x \frac{x ^{9} - 1}{x - 1} = \frac{x ^{10} - x}{x - 1}

. The only answer with this term is

\frac{x ^{10} - x}{x - 1} + 45 a

.

Final answer

D