To find the value of [tex]\( k \)[/tex], we need to use the given information about the list of values and their median.
Given the list of numbers in ascending order:
[tex]\[ 5, k + 3, 2k + 1, 3k - 2, 26, \text{ and } 31 \][/tex]
The median of a sorted list with an even number of elements (in this case, 6 elements) is the average of the two middle numbers. So, the median is the average of the 3rd and 4th numbers in the sorted list.
The given median is 17, so we can write the equation for the median as:
[tex]\[ \text{Median} = \frac{(2k + 1) + (3k - 2)}{2} = 17 \][/tex]
Let's solve this step-by-step:
1. Combine the terms inside the numerator:
[tex]\[ (2k + 1) + (3k - 2) = 2k + 3k + 1 - 2 = 5k - 1 \][/tex]
2. Set up the equation for the median and solve for [tex]\( k \)[/tex]:
[tex]\[ \frac{5k - 1}{2} = 17 \][/tex]
3. Multiply both sides by 2 to clear the fraction:
[tex]\[ 5k - 1 = 34 \][/tex]
4. Add 1 to both sides to isolate the term with [tex]\( k \)[/tex]:
[tex]\[ 5k = 35 \][/tex]
5. Divide both sides by 5 to solve for [tex]\( k \)[/tex]:
[tex]\[ k = 7 \][/tex]
So, the value of [tex]\( k \)[/tex] is [tex]\( \boxed{7} \)[/tex].
