一个等比数列的第三项为 $50$，第五项为 $12.5$。

已知该数列的公比为正，求其无穷和。

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。

第五项比第三项多乘两次公比。由两项之比确定正公比，再从第三项退回首项。

If $a$ is the first term and $r>0$ is the common ratio, then $ar^2=50$ and $ar^4=12.5$. Dividing gives $r^2=12.5/50=1/4$, so $r=1/2$ and $a=50/(1/4)=200$. Since $|r|<1$, the sum to infinity is $200/(1-1/2)=400$.

第三项的平方除以第五项，公比的幂次正好抵消，可先直接求首项。

Write the third and fifth terms as $ar^2=50$ and $ar^4=12.5$. Dividing the square of the first equality by the second gives

Then $r^2=50/200=1/4$. Since $r>0$, $r=1/2$, which has absolute value less than $1$. Hence the sum to infinity is