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 ways of x is

• A. 26
• B. 32
• C. 40
• D. 38

Answer 1

The correct option is C 40

The given integers are 4, 4, 4, 8, 10, 20 and x
Consider (a) x < 4
Mean $=\frac{50+x}{7},$ Median = 4, Mode = 4
If mean, median and mode are in AP then Here mean = 4
$\frac{50+x}{7}=4\Rightarrow x=-22$
Consider (b) 4<x<8
Mean $=\frac{50+x}{7},\frac{54}{7}<Mean<\frac{58}{7}$
Median = x
Mode = 4
As these are in AP Mean = 8
$\therefore x=6$
Consider (c) 8 < x
Median = 8
Mode = 4
As these are in AP, Mean = 12 i.e., x = 34
$\therefore$ can be -22, 6 or 34
The sum of these is 18, which is not given in the options.
If negative value of -22 is ignored then mean = median = mode = 4 is not considered.
$\therefore$ We have to select 40.
Answer in the given choices.