If a,b,c are in geometric progression and a,2b,3c are in arithmetic progression, then what is the common ratio r such that `0ltrlt1` ?

Options

A. `1//2`
B. `1//3`
C. `2//3`
D. none of these

Explanation

To solve the problem, we need to find the common ratio \( r \) such that \( 0 < r < 1 \), given that \( a, b, c \) are in geometric progression and \( a, 2b, 3c \) are in arithmetic progression.

### Step-by-Step Solution:

1. **Define the terms of the geometric progression**:
Since \( a, b, c \) are in geometric progression, we can express \( b \) and \( c \) in terms of \( a \) and the common ratio \( r \):
\[ b = ar \quad \text{and} \quad c = ar^2 \]

2. **Set up the arithmetic progression condition**:
We know that \( a, 2b, 3c \) are in arithmetic progression. For three terms \( x, y, z \) to be in arithmetic progression, the condition is:
\[ 2y = x + z \]
Applying this to our terms:
\[ 2(2b) = a + 3c \]
Substituting \( b \) and \( c \):
\[ 4b = a + 3c \]
This becomes:
\[ 4(ar) = a + 3(ar^2) \]

3. **Substituting and simplifying**:
Substitute \( b = ar \) and \( c = ar^2 \) into the equation:
\[ 4ar = a + 3ar^2 \]
Rearranging gives:
\[ 4ar - 3ar^2 - a = 0 \]

4. **Factor out \( a \)**:
Assuming \( a \neq 0 \), we can factor out \( a \):
\[ a(4r - 3r^2 - 1) = 0 \]
Thus, we need to solve:
\[ 4r - 3r^2 - 1 = 0 \]

5. **Rearranging into standard quadratic form**:
Rearranging gives us:
\[ 3r^2 - 4r + 1 = 0 \]

6. **Applying the quadratic formula**:
The quadratic formula is given by:
\[ r = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \]
Here, \( a = 3 \), \( b = -4 \), and \( c = 1 \):
\[ r = \frac{4 \pm \sqrt{(-4)^2 - 4 \cdot 3 \cdot 1}}{2 \cdot 3} \]
Simplifying the discriminant:
\[ r = \frac{4 \pm \sqrt{16 - 12}}{6} = \frac{4 \pm \sqrt{4}}{6} = \frac{4 \pm 2}{6} \]
This gives us two potential solutions:
\[ r = \frac{6}{6} = 1 \quad \text{and} \quad r = \frac{2}{6} = \frac{1}{3} \]

7. **Selecting the valid solution**:
Since we need \( 0 < r < 1 \), we discard \( r = 1 \) and keep:
\[ r = \frac{1}{3} \]

### Final Answer:
The common ratio \( r \) is \( \frac{1}{3} \).