一个等比数列与一个等差数列的首项都等于 $a$。

等比数列的第三项等于等差数列的第二项。

等比数列的第五项等于等差数列的第六项。

已知所有项都为正数且不全相等，用 $a$ 表示等差数列的前二十项的和。

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。

用等比数列的公比和等差数列的公差写两个相等条件。消去公差后，将公比的平方视为未知量。

Let $r$ be the common ratio and $d$ the common difference. The conditions give

Substituting $d=a(r^2-1)$ in the second equation and dividing by $a>0$ gives

Positive terms imply $r>0$. The value $r=1$ would give $d=0$ and all terms equal. Hence $r=2$ and $d=3a$. The required sum is $\frac{20}{2}(2a+19(3a))=590a$.

把第三项条件平方，再利用第五项条件，便能消去公比，直接求公差。

Let $r$ and $d$ be the common ratio and common difference. Then $ar^2=a+d$ and $ar^4=a+5d$. Squaring the first equation and using the second gives

If $d=0$, then $r^2=1$. Since all terms are positive, $r=1$, making all terms equal, contrary to the condition. Hence $d=3a$ and