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Algebra Difficulty 3.9 AMC 10/12 Find the answer China

Suppose that geometric sequence {an}\{a_n\} satisfies a1−a2=3a_1 - a_2 = 3, a1−a3=2a_1 - a_3 = 2. Then the common ratio of {an}\{a_n\} is ________.

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

Solution

Let the common ratio of {an}\{a_n\} be qq. Then
a1(1−q)=a1−a2=3,a1(1−q2)=a1−a3=2, \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=a1(1−q2)a1(1−q)=231 + q = \frac{a_1(1 - q^2)}{a_1(1 - q)} = \frac{2}{3}. Therefore, q=−13q = -\frac{1}{3}.
□\square

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