1950 AMC 12 第 5 题

Question

在

8

和

5832

之间插入五个等比中项后，所得等比数列的第五项是：

If five geometric means are inserted between

8

and

5832,

the fifth term in the geometric series is:

648

832

1168

1944

以上答案均不正确

None of these answers

插入五个等比中项后，两个已知数分别是第一项和第七项

Inserting five means makes the given numbers the first and seventh terms

若公比为

q

，则

8q^6=5832

If the common ratio is

q,

then

8q^6=5832

从第一项

8

到第七项

5832

需要连续乘六次公比。因此，

q^6=\frac{5832}{8}=729=3^6\text{，}

所以正公比为

q=3

。第五项是

8q^4=8\cdot3^4=648\text{。}

因此，正确答案是 A。

There are six common-ratio steps from the first term

8

to the seventh term

5832.

Thus

q^6=\frac{5832}{8}=729=3^6,

so the positive common ratio is

q=3.

The fifth term is

8q^4=8\cdot3^4=648.

Thus, the correct answer is A.

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