1971 AMC 12 第 13 题

Question

将

(1.0025)^{10}

的值精确到小数点后

5

位，则小数点后第五位数字是：

If

(1.0025)^{10}

is evaluated correct to

5

decimal places, then the digit in the fifth decimal place is:

0

1

2

5

8

写成

1.0025=1+0.0025

Write

1.0025=1+0.0025

在二项式展开中，保留足以影响小数点后第五位的项

In the binomial expansion, retain terms large enough to affect the fifth decimal place

二项式展开给出

\begin{gathered} (1+0.0025)^{10}\\ =1+10(0.0025)\\ \quad+45(0.0025)^2\\ \quad+120(0.0025)^3+\cdots\\ =1.025283125\ldots \end{gathered}

四舍五入后为

1.02528

。小数点后第五位数字是

8

。

因此，正确答案为 E。

The binomial expansion gives

\begin{gathered} (1+0.0025)^{10}\\ =1+10(0.0025)\\ \quad+45(0.0025)^2\\ \quad+120(0.0025)^3+\cdots\\ =1.025283125\ldots \end{gathered}

which rounds to

1.02528.

The fifth decimal digit is

8.

Therefore, the correct answer is E.

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