AMC12 2009 B

AMC12 2009 B · Q12

AMC12 2009 B · Q12. It mainly tests Sequences & recursion (algebra).

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

一个实数等比数列的第 5 项和第 8 项分别是 $7!$ 和 $8!$。首项是多少？

(A) 60 60

(B) 75 75

(D) 225 225

(E) 315 315

Answer

Correct choice: (E)

正确答案：（E）

Solution

Let the $n$th term of the series be $ar^{n-1}$. Because \[\frac {8!}{7!} = \frac {ar^7}{ar^4} = r^3 = 8,\] it follows that $r = 2$ and the first term is $a = \frac {7!}{r^4} = \frac {7!}{16} = \boxed{315}$. The answer is $\mathrm{(E)}$.

设该数列的第 $n$ 项为 $ar^{n-1}$。因为 \[\frac {8!}{7!} = \frac {ar^7}{ar^4} = r^3 = 8,\] 可得 $r = 2$，首项为 $a = \frac {7!}{r^4} = \frac {7!}{16} = \boxed{315}$。答案是 $\mathrm{(E)}$。

Topics

AL.012 Sequences & recursion (algebra)