2009 AMC 12B 第 12 题

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

第八项与第五项的比是

r^3

。

The ratio of the eighth term to the fifth term is

r^3

\dfrac{8!}{7!} = 8 = r^3

，所以

r = 2

；再用

7!

除以

r^4

。

\dfrac{8!}{7!} = 8 = r^3,

so

r = 2

; then divide

7!

by

r^4

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.

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