Discussion

Question:

In a list of 7 integers, one integer, denoted as x is unknown. The other six integers are 20, 4, 10, 4, 8, and 4. If the mean, median, and mode of these seven integers are arranged in increasing order, they form an arithmetic progression. The sum of all possible values of x is

Explanation:

It can be observed that, irrespective of the value of x, mode of these numbers will be 4.

Now, the median of these numbers will depend on the value of x,

If x < 4 then the median of these seven numbers will be 4.

Now, as the mode is 4, the median cannot be 4.

(the question states that mean, median and mode are arranged in ascending order.)

Hence x cannot be less than 4.

Now,

If 4 < x ≤ 8

the median will be x & the mean will be,

$\frac{50+x}{7}$

Now $\frac{50+x}{7}$ > 7 and x ≤ 8

∴ 4, x and $\frac{50+x}{7}$ will form an AP only if $\frac{50+x}{7}$ > x

∴ $\frac{50+x}{7}$ - x = x - 4

∴ $\frac{50+x}{7}$ + 4 = 2x

∴ x = 6

Hence, x = 6 is a possible answer.

Now, if x > 8 then median will be 8 & mean will be 

$\frac{50+x}{7}$

Now, if x > 8 then $\frac{50+x}{7}$ is greater than 8.

∴ Increasing order of mean, median and mode will be,

4, 8, $\frac{50+x}{7}$

Now, they are in A.P.

∴ 8 - 4 = $\frac{50+x}{7}$ - 8

∴ 12 = $\frac{50+x}{7}$

∴ x = 12 × 7 – 50

∴ x = 34

Hence, sum of all possible values of x = 6 + 34 = 40

Hence, option (e).

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