A finite arithmetic sequence has 7 terms, and the first term is `3/4`. What is the difference between the mean and the median of the 7 terms?

Options

A. 0
B. `3/4`
C. `4/3`
D. 3

Answer

To solve the problem, we need to find the difference between the mean and the median of a finite arithmetic sequence with 7 terms, where the first term is \( \frac{3}{4} \).

### Step 1: Define the terms of the arithmetic sequence
Let the first term of the arithmetic sequence be \( a = \frac{3}{4} \) and let \( d \) be the common difference. The 7 terms of the arithmetic sequence can be expressed as:
- First term: \( a = \frac{3}{4} \)
- Second term: \( a + d \)
- Third term: \( a + 2d \)
- Fourth term: \( a + 3d \)
- Fifth term: \( a + 4d \)
- Sixth term: \( a + 5d \)
- Seventh term: \( a + 6d \)

Thus, the 7 terms are:
\[ \frac{3}{4}, \frac{3}{4} + d, \frac{3}{4} + 2d, \frac{3}{4} + 3d, \frac{3}{4} + 4d, \frac{3}{4} + 5d, \frac{3}{4} + 6d \]

### Step 2: Calculate the median
Since there are 7 terms (an odd number), the median is the middle term, which is the 4th term:
\[ \text{Median} = a + 3d = \frac{3}{4} + 3d \]

### Step 3: Calculate the mean
The mean of the arithmetic sequence is calculated by summing all the terms and dividing by the number of terms (7):
\[ \text{Mean} = \frac{\left(\frac{3}{4} + \left(\frac{3}{4} + d\right) + \left(\frac{3}{4} + 2d\right) + \left(\frac{3}{4} + 3d\right) + \left(\frac{3}{4} + 4d\right) + \left(\frac{3}{4} + 5d\right) + \left(\frac{3}{4} + 6d\right)\right)}{7} \]
This simplifies to:
\[ \text{Mean} = \frac{7 \cdot \frac{3}{4} + (0 + 1 + 2 + 3 + 4 + 5 + 6)d}{7} \]
Calculating the sum of the integers:
\[ 0 + 1 + 2 + 3 + 4 + 5 + 6 = 21 \]
Thus,
\[ \text{Mean} = \frac{7 \cdot \frac{3}{4} + 21d}{7} = \frac{3}{4} + 3d \]

### Step 4: Find the difference between mean and median
Now we can find the difference between the mean and the median:
\[ \text{Difference} = \text{Mean} - \text{Median} = \left(\frac{3}{4} + 3d\right) - \left(\frac{3}{4} + 3d\right) = 0 \]

### Final Answer
The difference between the mean and the median of the 7 terms is \( 0 \).

---