一个等差数列的第 $7$ 项为 $77$。该数列的前 $n$ 项和记为 $S_n$，已知 $S_{11}=880$。

(a) 求该数列的第 $20$ 项。

(b) 求使 $S_N<50$ 的最小项数 $N$。

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。

前 $11$ 项的平均数是中间的第 $6$ 项。比较第 $6$ 和第 $7$ 项即可求公差。

For an arithmetic progression, $S_{11}=11(a+5d)$, so the sixth term is $80$. The seventh is $77$, giving $d=-3$ and $a=80-5(-3)=95$. The twentieth term is $95+19(-3)=38$.

把求和公式写成关于 $N$ 的二次不等式。先排除小于 $1$ 的一支，再取符合严格不等式的最小整数。

The inequality $S_N<50$ is

The roots are $(193\pm\sqrt{36049})/6$, approximately $0.522$ and $63.811$. The quadratic is positive outside the roots, so the smallest positive integer that works is $N=64$. In particular, $S_{63}=126$ and $S_{64}=32$.