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

If these are in A.P. => Mean = 4

=> $$\frac{50 + x}{7} = 4$$

=> $$50 + x = 28$$

=> $$x = 28 - 50 = -22$$

=> It is not possible

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 = $$6 + 34 = 40$$