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111學測數學A考科-04


<單選題>設等差數列 \( \langle a_n \rangle \) 之首項 \( a_1 \) 與公差 \( d \) 皆為正數,且 \( \log a_1 \cdot \log a_3 \cdot \log a_6 \) 依序也成等差數列。試選出數列 \( \log a_1 \cdot \log a_3 \cdot \log a_6 \) 的公差。
(1) \( \log d \) (2) \( \log \frac{2}{3} \) (3) \( \log \frac{3}{2} \) (4) \( \log 2d \) (5) \( \log 3d \)

答案

由 \( \log a_1 + \log a_6 = 2 \log a_3 \) 得 \( a_1(a_1+5d) = (a_1+2d)^2 \Rightarrow a_1d=4d^2 \Rightarrow a_1=4d \)
公差 = \( \log a_3 - \log a_1 = \log \frac{a_1+2d}{a_1} = \log \frac{6d}{4d} = \log \frac{3}{2} \)
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