Question

一个实数等比数列的第五项和第八项分别是 $7!$ 和 $8!$ 。首项是多少？

The fifth and eighth terms of a geometric sequence of real numbers are $7!$ and $8!$ respectively. What is the first term?

$60$

$75$

$120$

$225$

$315$

第八项除以第五项得 $r^3 = \dfrac{8!}{7!} = 8$ ，所以 $r = 2$ 。

第五项为 $a r^4 = 7!$ ，所以 $a = \dfrac{7!}{16} = \dfrac{5040}{16} = 315$ 。

所以正确答案是 E。

The eighth term divided by the fifth term is $r^3 = \dfrac{8!}{7!} = 8,$ so $r = 2.$

The fifth term is $a r^4 = 7!,$ so $a = \dfrac{7!}{16} = \dfrac{5040}{16} = 315.$

Thus, the correct answer is E.