一个等比数列的第一、第二和第三项分别等于一个等差数列的第一、第四和第六项。两个数列的首项均非零。等比数列的公比为 $r$。

(a) 证明

(b) 已知等比数列的无穷和为 $54$。

求等差数列的公差。

Answer all questions. You should use a calculator where appropriate.

You must show all necessary working clearly; no marks will be given for unsupported answers from a calculator.

Give non-exact numerical answers correct to 3 significant figures, or 1 decimal place for angles in degrees, unless a different level of accuracy is specified in the question.

请回答所有题目。适当时应使用计算器。必须清楚展示所有必要的计算过程；仅给出计算器答案而没有过程，不予评分。除非题目另有精度要求，非精确数值答案保留 3 位有效数字，以度为单位的角度保留 1 位小数。

用同一个首项表示两列数。等差数列的第 $4$、$6$ 项分别比首项多 $3d$、$5d$，联立后消去公差。

Let the common first term be $u\ne0$ and the arithmetic difference be $d$. The conditions give

Hence $u(r-1)=3d$ and $u(r^2-1)=5d$. Multiplying the first by $5$ and the second by $3$ gives $5u(r-1)=3u(r^2-1)$. Dividing by the nonzero $u$ and rearranging gives $3r^2-5r+2=0$.

先从 (a) 的两个公比中选出可以求无穷和的一个。再用总和求首项，最后回到等差关系求公差。

The equation in (a) factors as $(3r-2)(r-1)=0$. Only $r=2/3$ satisfies $|r|<1$. From $u/(1-r)=54$, the first term is $u=18$. Then $3d=u(r-1)=18(-1/3)=-6$, giving $d=-2$.