If `a,b,c` be three positive numbers in `A.P.` and `E=(a+8b)/(2b-a)+(8b+c)/(2b-c)`, then a value of `E` can be

Options

A. `16`
B. `15`
C. `17`
D. `21`

Explanation

To solve the problem, we start with the given expression for \( E \) and the fact that \( a, b, c \) are in arithmetic progression (A.P.). This means that \( 2b = a + c \).

### Step-by-Step Solution:

1. **Substituting A.P. Condition**:
Since \( a, b, c \) are in A.P., we can express \( c \) in terms of \( a \) and \( b \):
\[ c = 2b - a \]

2. **Substituting \( c \) into \( E \)**:
The expression for \( E \) is given as:
\[ E = \frac{a + 8b}{2b - a} + \frac{8b + c}{2b - c} \]
Now, substitute \( c = 2b - a \) into the expression:
\[ E = \frac{a + 8b}{2b - a} + \frac{8b + (2b - a)}{2b - (2b - a)} \]
Simplifying the second term:
\[ E = \frac{a + 8b}{2b - a} + \frac{8b + 2b - a}{a} \]
\[ E = \frac{a + 8b}{2b - a} + \frac{10b - a}{a} \]

3. **Finding a Common Denominator**:
To combine the fractions, we find a common denominator:
\[ E = \frac{(a + 8b)a + (10b - a)(2b - a)}{a(2b - a)} \]

4. **Expanding the Numerator**:
Expanding the numerator:
\[ = \frac{a^2 + 8ab + (20b^2 - 10ab - 2a^2 + a^2)}{a(2b - a)} \]
\[ = \frac{(20b^2 - 2a^2 + 3ab)}{a(2b - a)} \]

5. **Analyzing the Expression**:
To analyze the expression further, we can simplify it further or substitute specific values for \( a \) and \( b \) to find the minimum value of \( E \).

6. **Using AM-GM Inequality**:
We can apply the AM-GM inequality to the terms \( \frac{a}{c} + \frac{c}{a} \):
\[ \frac{a}{c} + \frac{c}{a} \geq 2 \]
Thus, we can conclude that:
\[ E \geq 2 + 8 = 10 \]

7. **Finding a Specific Value**:
By testing values, we can find that the minimum value of \( E \) can be greater than 18. Therefore, we conclude that:
\[ E > 18 \]

### Conclusion:
The only option greater than 18 from the given choices is option 4.