一个数列的第一项、第二项和第三项分别为 $20$、$k$ 和 $k-5$。

(a) 已知这个数列是等差数列，求第 $30$ 项。

(b) 若改为已知这个数列是等比数列，求无穷和。

等差数列相邻两项的差相等。第二项到第三项已经给出了公差。

The common difference is $(k-5)-k=-5$. Hence the 30th term is

等比数列的相邻两项之比相同。先求出第二项，再算公比和无穷和。

The second term cannot be zero, since then the third term would also have to be zero. Equating the ratios gives

Thus $k=10$ and $r=\frac12$. Since $|r|<1$, the sum to infinity is $20/(1-\frac12)=40$.

也可以直接用公比表示第二项和第三项，从而省去先求 $k$ 的步骤。

With common ratio $r$, the second term is $k=20r$, and the third is $20r^2=k-5=20r-5$. Therefore

This gives $r=\frac12$. Since $|r|<1$, the sum to infinity is $20/(1-\frac12)=40$.