When the mean median and mode of the list

10,2,5,2,4,2,x

are arranged in increasing order, they form a non-constant arithmetic progression. What is the sum of all possible real values of x?

Thanks in advance,

Brad

Do you know what mode, mean and median are? If so, get an expression for each of these in terms of x. (The expression for the median isn't explicit - you have to divide it up into subcases depending on the value of x.)

The mode will always be 2 since there is no value x can take that will alter the mode. If x > 4, the median will be 4. If x < 2, the median will be 2. If 2 < x < 4, the median will be x. You should then be able to calculate the mean for each of these cases and from the mean calculate x.

Tom.

Using your hints, I've been able to get the answer 3.

Thanks, you've been a great help,

Brad

I think Brad has missed a solution. I get x=3 when the median is x, but what about if the median is 4?

When the mean median and mode of the list:
10.2,5,2,4.2,x are arranged in increasing order, they form a non constant arithmetic progression.What is the sum of all possible real values of x?
Thank you.
Maria Jose.

Notice in this case the mode makes sense if and only if x is one of 10.2, 5, 2 or 4.2. This means x will be the mode. Now, the mean is
(10.2+5+2+4.2+x)/5=(21.4+x)/5
and the median will be
5 if x > = 5
4.2 if x < = 4.2
The condition for non-trivial AP means x=10.2 or 2.

Now consider each case separately -
x=10.2: mode=10.2, median=5, mean = 6.32 - not an AP
x=2: mode=2, median = 5, mean= 23.4/5 =5.68 - not an AP

so there are no x's that satisfies the condition, hence the sum is taken over the empty set, and the answer is 0.

Kerwin