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

Integers = $$4,4,4,8,10,20,x$$

Clearly, irrespective of the value of $$x$$, Mode = $$4$$

Sum of above integers = $$4 + 4 + 4 + 8 + 10 + 20 + x$$

= $$50 + x$$

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

Case 1 : If $$x < 4$$

Median of $$x,4,4,4,8,10,20$$ = 4

Mode = 4

Since the median and mode of the dataset are equal, they cannot be arranged in increasing order.

So this case is rejected.

Case 2 : If $$4 < x < 8$$

Median of $$4,4,4,x,8,10,20$$ = $$x$$

Mode = 4

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

=> $$\frac{54}{7} < Mean < \frac{58}{7}$$

As these are in AP => $$x = 6$$ and Mean = $$8$$

Case 3 : If $$x > 8$$

Mean = $$\frac{50 + x}{7} > \frac{58}{7}$$

Median of $$4,4,4,8,x,10,20$$ = $$8$$

Mode = $$4$$

As these are in AP, => Mean = $$12$$

=> $$\frac{50 + x}{7} = 12$$

=> $$50 + x = 84$$

=> $$x = 84 - 50 = 34$$

$$\therefore$$ Sum of all possible values of x is = $$6 + 34 = 40$$