一个等比数列的前两项之和为 $15$，无穷和为 $\frac{125}{7}$，且公比为负数。

求该数列的第三项。

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。

把前两项和与无穷和写成首项和公比的式子。两式相除消去首项，再按题意取负公比。

Let the first term be $a$ and the ratio be $r$. Then $a(1+r)=15$ and $a/(1-r)=125/7$. Dividing gives

Thus $r^2=4/25$, so $r=-2/5$. It follows that $a=15/(1-2/5)=25$, and the third term is $ar^2=4$.

由前两项和把公比表示成首项的式子，再代入无穷和。求出首项的两个候选值后，用公比为负这一条件选择。

Let the first term be $a$ and the common ratio be $r$. Since $a(1+r)=15$, we have $a\ne0$ and $r=15/a-1$. Substituting in the sum to infinity gives

As the progression converges, $r\ne1$, so $2a-15\ne0$. Clearing the denominator gives

If $a=25$, then $r=-2/5$. If $a=75/7$, then $r=2/5$, which is excluded because the ratio is negative. Hence the third term is $25(-2/5)^2=4$.