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 mode is always $2,$ and the mean is $\dfrac{25 + x}{7}.$ For the values to form a non-constant arithmetic progression we examine the median.

If $x = 3,$ the sorted list is $2, 2, 2, 3, 4, 5, 10,$ with median $3$ and mean $4,$ giving the progression $2, 3, 4.$

If $x = 17,$ the sorted list is $2, 2, 2, 4, 5, 10, 17,$ with median $4$ and mean $6,$ giving the progression $2, 4, 6.$

No other value of $x$ works, so the sum of all possible values is $3 + 17 = 20.$

Thus, the correct answer is E.

2000 AMC 10 Problem 23

Solution:

Problem 23 in Other Years