If a, b, c are in A.P., then `(a)/(bc), (1)/(c), (2)/(b)` are also in A.P.

Answer

To determine whether the numbers \(\frac{a}{bc}, \frac{1}{c}, \frac{2}{b}\) are in arithmetic progression (A.P.) given that \(a, b, c\) are in A.P., we can follow these steps:

### Step 1: Understand the condition for A.P.
For three numbers \(x, y, z\) to be in A.P., the following condition must hold:
\[ 2y = x + z \]

### Step 2: Set the terms
Let:
- \(x = \frac{a}{bc}\)
- \(y = \frac{1}{c}\)
- \(z = \frac{2}{b}\)

### Step 3: Apply the A.P. condition
We need to check if:
\[ 2 \cdot \frac{1}{c} = \frac{a}{bc} + \frac{2}{b} \]

### Step 4: Simplify the left-hand side
Calculating the left-hand side:
\[ 2 \cdot \frac{1}{c} = \frac{2}{c} \]

### Step 5: Simplify the right-hand side
Now, we simplify the right-hand side:
\[ \frac{a}{bc} + \frac{2}{b} = \frac{a + 2c}{bc} \]

### Step 6: Set the two sides equal
Now we equate both sides:
\[ \frac{2}{c} = \frac{a + 2c}{bc} \]

### Step 7: Cross-multiply to eliminate the fractions
Cross-multiplying gives:
\[ 2bc = c(a + 2c) \]

### Step 8: Simplify the equation
Assuming \(c \neq 0\), we can divide both sides by \(c\):
\[ 2b = a + 2c \]

### Step 9: Use the A.P. condition of \(a, b, c\)
Since \(a, b, c\) are in A.P., we have:
\[ 2b = a + c \]

### Step 10: Compare the two equations
From our derived equation \(2b = a + 2c\) and the A.P. condition \(2b = a + c\), we can see that:
\[ a + 2c = a + c \]
This implies:
\[ c = 0 \]

### Conclusion
Since \(c = 0\) is not a valid case in the context of A.P. (as it would make the fractions undefined), we conclude that the statement is false. Therefore, \(\frac{a}{bc}, \frac{1}{c}, \frac{2}{b}\) are not in A.P.