一个等差数列的前三项为 $\frac{p^2}{6}$、$2p-6$ 和 $p$。

(a) 已知该数列的公差不为零，求 $p$ 的值。

(b) 使用这个值，求首两项为 $\frac{p^2}{6}$ 和 $2p-6$ 的等比数列的无穷和。

等差数列的中项等于首尾两项的平均。解出 $p$ 后还要检查公差不为零。

The arithmetic condition gives

For $p=6$, all three terms are $6$, so the common difference is zero. Hence $p=12$.

以公差为未知数。先由第二、三项表达 $p$，再把首项写成公差的式子。

Let the nonzero common difference be $d$. From the second and third terms,

The first term is $p-2d=6-3d$, so

Since $d\ne0$, $d=-6$ and $p=6-(-6)=12$.

用 (a) 的 $p$ 求等比数列的首项和公比。

The first two terms are $24$ and $18$, so the common ratio is $3/4$. Since $|3/4|<1$, the sum to infinity is $24/(1-3/4)=96$.