1950 AMC 12 第 43 题

Question

无穷级数

\dfrac17+\dfrac{2}{7^2}+\dfrac{1}{7^3}+\dfrac{2}{7^4}+\cdots

的和为：

The sum to infinity of

\dfrac17+\dfrac{2}{7^2}+\dfrac{1}{7^3}+\dfrac{2}{7^4}+\cdots

is:

\dfrac15

\dfrac1{24}

\dfrac5{48}

\dfrac1{16}

以上答案均不正确

None of these

将级数中相邻的两项分为一组

Group the series into consecutive pairs of terms

每一组都是前一组的

\frac{1}{49}

Each pair is

\frac{1}{49}

times the preceding pair

将相邻两项分组后，得到一个等比级数，其首项为

\frac17+\frac2{49}=\frac9{49}\text{，}

公比为

\tfrac1{49}

。因此，级数之和为

\frac{\frac9{49}}{1-\frac1{49}} =\frac9{48} =\frac3{16}\text{。}

这个值不在选项 A 至 D 中。

因此，正确答案是 E。

Grouping consecutive terms gives a geometric series whose first grouped term is

\frac17+\frac2{49}=\frac9{49}

and whose ratio is

\tfrac1{49}.

Hence the sum is

\frac{\frac9{49}}{1-\frac1{49}} =\frac9{48} =\frac3{16}.

This is not among choices A through D.

Thus, the correct answer is E.

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