Sure! Let's solve the problem step by step.
We are given a list of six terms: [tex]\(5, k+3, 2k+1, 3k-2, 26, \text{and} 31\)[/tex] in ascending order. The median of these terms is given as 17, and we need to find the value of [tex]\(k\)[/tex].
Since the list has 6 terms, the median will be the average of the 3rd and 4th terms in this ordered list.
1. Identify the 3rd and 4th terms in the list:
- The 3rd term is [tex]\(2k + 1\)[/tex]
- The 4th term is [tex]\(3k - 2\)[/tex]
2. Calculate the median of the 3rd and 4th terms:
[tex]\[
\text{Median} = \frac{(2k + 1) + (3k - 2)}{2}
\][/tex]
3. Set this equal to the given median value, which is 17:
[tex]\[
\frac{(2k + 1) + (3k - 2)}{2} = 17
\][/tex]
4. Simplify the equation:
[tex]\[
\frac{2k + 3k + 1 - 2}{2} = 17
\][/tex]
[tex]\[
\frac{5k - 1}{2} = 17
\][/tex]
5. Solve for [tex]\(k\)[/tex]:
First, multiply both sides by 2 to clear the fraction:
[tex]\[
5k - 1 = 34
\][/tex]
Next, add 1 to both sides:
[tex]\[
5k = 35
\][/tex]
Finally, divide by 5:
[tex]\[
k = 7
\][/tex]
Therefore, the value of [tex]\(k\)[/tex] is [tex]\(7\)[/tex].
