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There is a list of seven integers in which one integer is unknown and the other six integers are given as 20,5,12,5,9,5. If the mean, median and mode of these seven integers are in arithmetic progression. Then the sum of all positive value of unknown integers is


Solution

Let us arrange the six known integers in ascending order : {5,5,5,9,12,20} and unknown integer =x.
As 5 occurs 3 times irrespective of value of x , mode of this list is 5
and Mean=5+5+5+9+12+20+x7=56+x7 
The possible value of median are 
(i)   5          if x5 
(ii) x          if 5<x9
(iii)  9          if x>9

Here, Median 5, as Mode =5x=21
Not possible 

Case-I:  5<x9
Mean + Mode = 2Median
x+567+5=2xx=7

Case-II: x>9
Mean + Mode = 2Median
x+567+5=18x=35

So, sum of all possible value of x=7+35=42.

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