Sure, let's break down the problem step by step to find the deleted number and then address the subsequent part of the question.
### Part 1: Finding the Deleted Number
1. Determine the total sum of the original 6 numbers:
- If the mean of the 6 numbers is 20, the total sum of these 6 numbers is given by:
[tex]\[
\text{Total sum of 6 numbers} = \text{mean} \times \text{number of elements} = 20 \times 6 = 120
\][/tex]
2. Determine the total sum after one number is deleted:
- After deleting one number, we have 5 numbers remaining.
- The mean of these 5 numbers is 15. Therefore, the total sum of these 5 numbers is:
[tex]\[
\text{Total sum of 5 numbers} = \text{mean} \times \text{number of elements} = 15 \times 5 = 75
\][/tex]
3. Calculate the deleted number:
- The deleted number is the difference between the total sum of the original numbers and the total sum of the remaining numbers:
[tex]\[
\text{Deleted number} = 120 - 75 = 45
\][/tex]
Therefore, the deleted number is [tex]\(\boxed{45}\)[/tex].
### Part 2: Calculating the Third Quartile (Q3)
Given the sequence in ascending order: [tex]\(1, 5, 7, 2x-4, x+7, 2x-1\)[/tex] and [tex]\(3x+2\)[/tex], and knowing that the third quartile [tex]\(Q3\)[/tex] is 15:
1. Determine the position of the third quartile for 7 numbers:
- For a set of 7 numbers, [tex]\(Q3\)[/tex] is the value of the number at the [tex]\( \left(\frac{3(n+1)}{4}\right) \)[/tex]th position:
[tex]\[
\text{Position of } Q3 = \left(\frac{3(7+1)}{4}\right) = \left(\frac{24}{4}\right) = 6
\][/tex]
- Therefore, the third quartile is the value of the 6th element in the ordered sequence.
2. Interpretation:
- We know from the given that [tex]\(Q3 = 15\)[/tex].
- Hence, the value of the 6th element (when sorted) is 15.
- The 6th element in the ordered sequence corresponds to [tex]\(2x-1\)[/tex].
3. Solve for [tex]\(x\)[/tex]:
[tex]\[
2x - 1 = 15
\][/tex]
[tex]\[
2x = 16
\][/tex]
[tex]\[
x = 8
\][/tex]
Thus, the value of [tex]\(x\)[/tex] that satisfies the given conditions is [tex]\( \boxed{8} \)[/tex].
