If the first three terms of an A.P. are b, c and 4b, then find the ratio of b and c.

Answer

To solve the problem, we need to find the ratio of \( b \) and \( c \) given that the first three terms of an arithmetic progression (A.P.) are \( b \), \( c \), and \( 4b \).

### Step-by-Step Solution:

1. **Understanding the A.P. Condition**:
Since \( b \), \( c \), and \( 4b \) are in A.P., the common difference between the terms must be equal. This means:
\[ c - b = 4b - c \]

2. **Setting Up the Equation**:
From the equation above, we can rearrange it:
\[ c - b = 4b - c \]
Adding \( c \) to both sides gives:
\[ c + c - b = 4b \]
Simplifying this, we have:
\[ 2c - b = 4b \]

3. **Rearranging the Equation**:
Now, let's isolate \( c \):
\[ 2c = 4b + b \]
\[ 2c = 5b \]

4. **Finding the Ratio**:
To find the ratio of \( b \) to \( c \), we divide both sides of the equation \( 2c = 5b \) by \( c \) and \( b \):
\[ \frac{b}{c} = \frac{2}{5} \]

5. **Final Answer**:
Therefore, the ratio of \( b \) to \( c \) is:
\[ \frac{b}{c} = \frac{2}{5} \]