Let's go step by step to determine the sum of all positive values of [tex]\( X \)[/tex] that satisfy the given condition.
We have a data set consisting of six positive integers: [tex]\[ 1, 7, 5, 2, 5, X \][/tex]
Let’s denote the average (arithmetic mean) of these six numbers as [tex]\( A \)[/tex].
Given that the average is one of the values in the data set or [tex]\( X \)[/tex] itself, we need to find [tex]\( X \)[/tex] such that:
[tex]\[ A \in \{ 1, 7, 5, 2, 5, X \} \][/tex]
Firstly, the formula for the arithmetic mean of the six numbers is:
[tex]\[ A = \frac{1 + 7 + 5 + 2 + 5 + X}{6} \][/tex]
Let’s compute the known sum of the integers given in the data set, excluding [tex]\( X \)[/tex]:
[tex]\[ 1 + 7 + 5 + 2 + 5 = 20 \][/tex]
Thus, the formula for the mean becomes:
[tex]\[ A = \frac{20 + X}{6} \][/tex]
Since [tex]\( A \)[/tex] must be a positive integer from the set \{ 1, 7, 5, 2, 5, X \}, we compare with each possibility by substituting these values for [tex]\( A \)[/tex].
1. [tex]\( A = 1 \)[/tex]:
[tex]\[ 1 = \frac{20 + X}{6} \][/tex]
[tex]\[ 6 = 20 + X \][/tex]
[tex]\[ X = -14 \][/tex] (Not a positive integer)
2. [tex]\( A = 7 \)[/tex]:
[tex]\[ 7 = \frac{20 + X}{6} \][/tex]
[tex]\[ 42 = 20 + X \][/tex]
[tex]\[ X = 22 \][/tex] (Positive integer)
3. [tex]\( A = 5 \)[/tex]:
[tex]\[ 5 = \frac{20 + X}{6} \][/tex]
[tex]\[ 30 = 20 + X \][/tex]
[tex]\[ X = 10 \][/tex] (Positive integer)
4. [tex]\( A = 2 \)[/tex]:
[tex]\[ 2 = \frac{20 + X}{6} \][/tex]
[tex]\[ 12 = 20 + X \][/tex]
[tex]\[ X = -8 \][/tex] (Not a positive integer)
5. [tex]\( A = X \)[/tex]:
This implies that [tex]\( X = A \)[/tex], in other words, the candidate [tex]\( X \)[/tex] must be equal to any previous valid [tex]\( X \)[/tex].
Summarizing, the positive integer values for [tex]\( X \)[/tex] that satisfy [tex]\( A \)[/tex] being one of the values in the data set are [tex]\( X = 10 \)[/tex] and [tex]\( X = 22 \)[/tex].
Now, we sum these valid values of [tex]\( X \)[/tex]:
[tex]\[ 10 + 22 = 32 \][/tex]
Thus, the sum of all positive values of [tex]\( X \)[/tex] is [tex]\( \boxed{32} \)[/tex].
