Question

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

Answer 1

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 $\text{Mean}=\frac{5+5+5+9+12+20+x}{7}=\frac{56+x}{7}$
The possible value of median are
$\left(i\right)5$ if $x\le 5$
$\left(ii\right)x$ if $5<x\le 9$
$\left(iii\right)9$ if $x>9$

Here, Median $\ne 5,$ as Mode $=5\Rightarrow x=-21$
Not possible

Case-I: $5<x\le 9$
Mean + Mode = 2Median
$\Rightarrow \frac{x+56}{7}+5=2x\Rightarrow x=7$

Case-II: $x>9$
Mean + Mode = 2Median
$\Rightarrow \frac{x+56}{7}+5=18\Rightarrow x=35$

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