Question

When the mean, median, and mode of the list $10, 2, 5, 2, 4, 2, x$

are arranged in increasing order, they form a non-constant arithmetic progression. What is the sum of all possible real values of $x?$

$3$

$6$

$9$

$17$

$20$

Solution:

The six fixed numbers sum to $25,$ so the mean is $\dfrac{25 + x}{7},$ and the mode is $2.$ If $x \le 2,$ then $2$ is both median and mode, forcing a constant progression, so $x \gt 2.$

Case $2 \lt x \lt 4$ : the median is $x.$ Requiring $2, x, \dfrac{25 + x}{7}$ to form an arithmetic progression yields $x = 3$ as the only value in this range.

Case $x \ge 4$ : the median is $4,$ and the progression $2, 4, 6$ forces the mean to be $6,$ so $\dfrac{25 + x}{7} = 6,$ giving $x = 17.$

The sum of all possible values is $3 + 17 = 20.$

Thus, the correct answer is E.

2000 AMC 12 Problem 14

Solution:

Problem 14 in Other Years