To solve the problem, we need to find the value of N such that the mean of the numbers 8, 9, 11, 15, 17, 21, and N is equal to the median of these numbers.
### Step 1: Calculate the Mean
The mean of a set of numbers is calculated by dividing the sum of the numbers by the total count of the numbers.
1. The numbers we have are: 8, 9, 11, 15, 17, 21, and N.
2. The total count of numbers is 7 (including N).
3. The sum of the numbers is: \( 8 + 9 + 11 + 15 + 17 + 21 + N = 81 + N \).
4. Therefore, the mean is given by:
\[
\text{Mean} = \frac{81 + N}{7}
\]
### Step 2: Determine the Median
To find the median, we need to arrange the numbers in ascending order. The median is the middle value when the numbers are sorted.
1. If N is less than 8, the sorted order will be: N, 8, 9, 11, 15, 17, 21, and the median will be the 4th number, which is 11.
2. If N is between 8 and 9, the sorted order will be: 8, N, 9, 11, 15, 17, 21, and the median will still be 11.
3. If N is between 9 and 11, the sorted order will be: 8, 9, N, 11, 15, 17, 21, and the median will still be 11.
4. If N is between 11 and 15, the sorted order will be: 8, 9, 11, N, 15, 17, 21, and the median will be N.
5. If N is between 15 and 17, the sorted order will be: 8, 9, 11, 15, N, 17, 21, and the median will be 15.
6. If N is between 17 and 21, the sorted order will be: 8, 9, 11, 15, 17, N, 21, and the median will be 15.
7. If N is greater than 21, the sorted order will be: 8, 9, 11, 15, 17, 21, N, and the median will be 15.
### Step 3: Set Mean Equal to Median
Now we can set the mean equal to the median based on the ranges of N we identified.
1. For \( N < 15 \):
\[
\frac{81 + N}{7} = 11
\]
Solving this:
\[
81 + N = 77 \implies N = -4 \quad (\text{not valid since } N \text{ must be positive})
\]
2. For \( N = 15 \):
\[
\frac{81 + 15}{7} = 13.714 \quad (\text{not equal to median})
\]
3. For \( N > 15 \) and \( N < 21 \):
\[
\frac{81 + N}{7} = 15
\]
Solving this:
\[
81 + N = 105 \implies N = 24 \quad (\text{not valid since } N < 21)
\]
4. For \( N \geq 21 \):
\[
\frac{81 + N}{7} = 15
\]
Solving this:
\[
81 + N = 105 \implies N = 24 \quad (\text{valid since } N > 21)
\]
### Conclusion
The value of N that satisfies the condition that the mean equals the median is:
\[
\boxed{24}
\]
