2021 AMC 10A Spring 第 4 题

Question

一辆小车沿山坡向下滚动，第一秒行驶

5

英寸，并且不断加速，使得之后每个连续的

1

秒时间段内，它都比前一个

1

秒时间段多行驶

7

英寸。小车用

30

秒到达山脚。它一共行驶了多少英寸？

A cart rolls down a hill, traveling

5

inches the first second and accelerating so that during each successive

1

-second time interval, it travels

7

inches more than during the previous

1

-second interval. The cart takes

30

seconds to reach the bottom of the hill. How far, in inches, does it travel?

215

360

2992

3195

3242

每秒行驶的距离组成一个等差数列。

The distances traveled each second form an arithmetic sequence

先求第

30

项，再用首项和末项的平均数求和。

Find the

30

th term, then use the average of the first and last terms

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每秒行驶的距离构成等差数列：

5, 5 + 7, 5 + 2 \cdot 7, \ldots

等差数列求和的标准公式为

n\cdot\dfrac{a_1+a_n}{2}\text{。}

已知项数为

30

，首项为

5

。末项为

5 + 29 \cdot 7 = 208\text{。}

把这些值代入公式，得到

30 \cdot \dfrac{5 + 208}{2} = 15 \cdot 213 = 3195\text{。}

所以正确答案是 D。

The distance travelled every second forms an arithmetic sequence:

5, 5 + 7, 5 + 2 \cdot 7, \ldots

The standard arithmetic-sequence sum formula is

n\cdot\dfrac{a_1+a_n}{2}.

We know the number of terms is

30

and the first term is

5.

The last term is

5 + 29 \cdot 7 = 208.

Plugging these values into the expression yields

30 \cdot \dfrac{5 + 208}{2} = 15 \cdot 213 = 3195.

Thus, D is the correct answer.

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