Three numbers form an increasing G.P. If the middle number is doubled, then the new numbers are in A.P. The common ratio of G.P. of

Options

A. `2 - sqrt(3)`
B. `2 + sqrt(3)`
C. `sqrt(3) - 2`
D. none of these

Answer

To solve the problem, let's denote the three numbers in the increasing geometric progression (G.P.) as \( a \), \( ar \), and \( ar^2 \), where \( a \) is the first term and \( r \) is the common ratio.

### Step 1: Define the G.P. terms
The three numbers in G.P. can be expressed as:
- First term: \( a \)
- Second term: \( ar \)
- Third term: \( ar^2 \)

### Step 2: Doubling the middle term
When the middle term \( ar \) is doubled, the new terms become:
- First term: \( a \)
- New second term: \( 2ar \)
- Third term: \( ar^2 \)

### Step 3: Setting up the condition for A.P.
For these new terms to be in arithmetic progression (A.P.), the condition is that the difference between the second and first term should be equal to the difference between the third and second term. This can be expressed mathematically as:
\[ 2ar - a = ar^2 - 2ar \]

### Step 4: Simplifying the equation
Now, simplify the equation:
\[ 2ar - a = ar^2 - 2ar \]
Combine like terms:
\[ 2ar - a + 2ar = ar^2 \]
This simplifies to:
\[ 4ar - a = ar^2 \]

### Step 5: Rearranging the equation
Rearranging gives:
\[ ar^2 - 4ar + a = 0 \]

### Step 6: Factoring the equation
We can factor out \( a \) (assuming \( a \neq 0 \)):
\[ a(r^2 - 4r + 1) = 0 \]
Since \( a \neq 0 \), we can focus on the quadratic:
\[ r^2 - 4r + 1 = 0 \]

### Step 7: Solving the quadratic equation
Using the quadratic formula \( r = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \):
Here, \( a = 1 \), \( b = -4 \), and \( c = 1 \):
\[ r = \frac{4 \pm \sqrt{(-4)^2 - 4 \cdot 1 \cdot 1}}{2 \cdot 1} \]
\[ r = \frac{4 \pm \sqrt{16 - 4}}{2} \]
\[ r = \frac{4 \pm \sqrt{12}}{2} \]
\[ r = \frac{4 \pm 2\sqrt{3}}{2} \]
\[ r = 2 \pm \sqrt{3} \]

### Final Answer
The common ratio \( r \) of the G.P. can be either \( 2 + \sqrt{3} \) or \( 2 - \sqrt{3} \).