In the sequence x,x+d, x+2d,x+3d , assume that x and d are positive integers . What is the difference between the arithmetic mean and the median of the numbers in the sequence ?

Options

A. 2x + 3d
B. `(2x + 3d ) / 2 `
C. `(x + d)/2`
D. 0

Answer

To solve the problem, we need to find the difference between the arithmetic mean and the median of the sequence \( x, x+d, x+2d, x+3d \).

### Step 1: Identify the terms of the sequence
The sequence consists of four terms:
1. \( x \)
2. \( x + d \)
3. \( x + 2d \)
4. \( x + 3d \)

### Step 2: Calculate the Arithmetic Mean
The arithmetic mean (AM) is calculated by taking the sum of all terms and dividing by the number of terms.

\[ \text{AM} = \frac{x + (x + d) + (x + 2d) + (x + 3d)}{4} \]

Calculating the sum:

\[ x + (x + d) + (x + 2d) + (x + 3d) = 4x + (0 + d + 2d + 3d) = 4x + 6d \]

Now, divide by the number of terms (which is 4):

\[ \text{AM} = \frac{4x + 6d}{4} = x + \frac{3d}{2} \]

### Step 3: Calculate the Median
Since there are four terms (an even number), the median is the average of the two middle terms. The middle terms in the sequence are \( x + d \) and \( x + 2d \).

Calculating the median:

\[ \text{Median} = \frac{(x + d) + (x + 2d)}{2} = \frac{2x + 3d}{2} = x + \frac{3d}{2} \]

### Step 4: Find the Difference Between the Arithmetic Mean and the Median
Now, we need to find the difference between the arithmetic mean and the median:

\[ \text{Difference} = \text{AM} - \text{Median} \]

Substituting the values we found:

\[ \text{Difference} = \left(x + \frac{3d}{2}\right) - \left(x + \frac{3d}{2}\right) = 0 \]

### Final Answer
The difference between the arithmetic mean and the median of the numbers in the sequence is \( 0 \).

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