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?

$215$

$360$

$2992$

$3195$

$3242$

答案：D

知识点：等差数列求和

难度评级：870

小提示：

每秒行驶的距离组成一个等差数列。

The distances traveled each second form an arithmetic sequence

大提示：

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

视频讲解：

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文字解答：

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.

2021 AMC 10A Spring 第 4 题