1964 AMC 12 第 6 题

Question

若

x

、

2x+2

、

3x+3

、

\ldots

构成等比数列，则第四项是：

If

x,

2x+2,

3x+3,

\ldots

are in geometric progression, the fourth term is:

-27

-13\dfrac12

12

13\dfrac12

27

三个连续非零项中，中间一项的平方等于相邻两项的乘积

For three consecutive nonzero terms, the middle term squared equals the product of its neighbors

求出

x

后，将第三项乘以公比

After finding

x,

multiply the third term by the common ratio

等比中项关系给出

(2x+2)^2=x(3x+3)\text{。}

按本试题对等比数列的公比约定，相邻项之比必须有定义，所以

x\ne-1

。两边除以

x+1

，得

4(x+1)=3x

，从而

x=-4

。前三项为

-4,-6,-9

，公比为

\frac{3}{2}

。第四项是

-9(\frac{3}{2})=-\frac{27}{2}

。

因此，正确答案是 B。

The geometric-mean relation gives

(2x+2)^2=x(3x+3).

Under the examination’s ratio convention for a geometric progression, the consecutive ratios must be defined, so

x\ne-1.

Dividing by

x+1

yields

4(x+1)=3x

and

x=-4.

The first three terms are

-4,-6,-9,

with ratio

\frac{3}{2}.

The fourth is

-9(\frac{3}{2})=-\frac{27}{2}.

Therefore, the correct answer is B.

其他年份的第 6 题