If a,b,c,d are distinct numbers such that : ` a + C = 2d` and `b+d = 2c` , which of the following statements must be ture ?
I. a cannot be the average of a,b,c,d.
II b can be the average of a,b,c,d .
III d can be the average of a,b,c,d.

Options

A. Only statement (I)
B. Only statement (II)
C. Only statement (III)
D. Both statements (I) and (II)

Answer

To solve the problem, we need to analyze the given equations and check the validity of each statement regarding the averages of the numbers \(a\), \(b\), \(c\), and \(d\).

### Step-by-Step Solution:

1. **Understand the Given Equations**:
We have two equations:
\[ a + c = 2d \quad \text{(1)} \]
\[ b + d = 2c \quad \text{(2)} \]

2. **Calculate the Average of \(a\), \(b\), \(c\), and \(d\)**:
The average \(A\) of the four numbers \(a\), \(b\), \(c\), and \(d\) is given by:
\[ A = \frac{a + b + c + d}{4} \]

3. **Substituting from the Given Equations**:
From equation (1), we can express \(a + c\) in terms of \(d\):
\[ a + c = 2d \implies c = 2d - a \quad \text{(3)} \]
From equation (2), we can express \(b + d\) in terms of \(c\):
\[ b + d = 2c \implies b = 2c - d \quad \text{(4)} \]

4. **Substituting \(c\) from Equation (3) into Equation (4)**:
Replace \(c\) in equation (4):
\[ b = 2(2d - a) - d = 4d - 2a - d = 3d - 2a \quad \text{(5)} \]

5. **Now Substitute \(b\) and \(c\) into the Average Formula**:
Substitute equations (3) and (5) into the average formula:
\[ A = \frac{a + (3d - 2a) + (2d - a) + d}{4} \]
Simplifying this:
\[ A = \frac{a + 3d - 2a + 2d - a + d}{4} = \frac{(3d + 2d + d) + (a - 2a - a)}{4} = \frac{6d - 2a}{4} = \frac{3d - a}{2} \]

6. **Analyzing Each Statement**:
- **Statement I**: \(a\) cannot be the average of \(a, b, c, d\).
- From our average calculation, \(A = \frac{3d - a}{2}\). For \(a\) to be the average, we would need \(a = \frac{3d - a}{2}\), which leads to \(2a = 3d - a\) or \(3a = 3d\) or \(a = d\). Since \(a\) and \(d\) are distinct, this statement is **true**.

- **Statement II**: \(b\) can be the average of \(a, b, c, d\).
- For \(b\) to be the average, we would need \(b = \frac{3d - a}{2}\). This is not guaranteed since \(b\) depends on \(d\) and \(a\) and does not necessarily equal the average. Thus, this statement is **false**.

- **Statement III**: \(d\) can be the average of \(a, b, c, d\).
- For \(d\) to be the average, we would need \(d = \frac{3d - a}{2}\). This leads to \(2d = 3d - a\) or \(d + a = 0\), which is not guaranteed since \(a\) and \(d\) are distinct. Thus, this statement is **false**.

### Conclusion:
Only Statement I is true. Therefore, the answer is that only Statement I must be true.