Question

从 $a$ 开始的五个连续正整数的平均数为 $b$ 。从 $b$ 开始的 $5$ 个连续整数的平均数是多少？

Five positive consecutive integers starting with $a$ have average $b.$ What is the average of $5$ consecutive integers that start with $b?$

$a+3$

$a+4$

$a+5$

$a+6$

$a+7$

从 $x$ 开始的五个连续整数的平均数是 $x+2$ 。

The average of five consecutive integers starting at $x$ is $x+2$

先对 $a$ 用这个规则，再对 $b$ 用一次。

Apply that rule first to $a$ , then to $b$

从 $x$ 开始的 $5$ 个连续整数的平均数为 $\dfrac{5x + 1 + 2 + 3 + 4}{5} = x + 2\text{。}$

因此，从 $a$ 开始的 $5$ 个连续整数的平均数为 $a + 2$ ，也就是 $b$ 。

从 $b$ 开始的 $5$ 个连续整数的平均数为 $b + 2 = a + 4\text{。}$

所以正确答案是 B。

Note that the average of $5$ consecutive numbers starting with $x$ is $\dfrac{5x + 1 + 2 + 3 + 4}{5} = x + 2.$

This means that the average of $5$ consecutive integers starting with $a$ is $a + 2,$ which we know is $b.$

Furthermore, the average of $5$ consecutive numbers starting with $b$ is $b + 2 = a + 4.$

Thus, B is the correct answer.