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Problem 718

AMC 10/12, early questions

Algebra Difficulty 3.9 Find the answer China Mathematical Competition · China

Suppose that geometric sequence $\{a_n\}$ satisfies $a_1 - a_2 = 3$ , $a_1 - a_3 = 2$ . Then the common ratio of $\{a_n\}$ is ________.

Your answer

A number or a short expression. Fractions can be typed as 3/2, and spacing doesn't matter.

Official solution

Let the common ratio of $\{a_n\}$ be $q$ . Then
$\begin{aligned} a_1(1-q) &= a_1 - a_2 = 3, \\ a_1(1-q^2) &= a_1 - a_3 = 2, \end{aligned}$
and thus $1 + q = \frac{a_1(1 - q^2)}{a_1(1 - q)} = \frac{2}{3}$ . Therefore, $q = -\frac{1}{3}$ .
$\square$

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