Does there exist a geometric progression containing 27,8 and 12 as three of its term ? If it exists, then how many such progressions are possible ?

Explanation

To determine whether there exists a geometric progression (GP) containing the terms 27, 8, and 12, we can follow these steps:

### Step 1: Assume the Terms
Assume that 27, 8, and 12 are the p-th, q-th, and r-th terms of a GP, respectively. Let the first term of the GP be \( A \) and the common ratio be \( r \).

### Step 2: Write the Terms in Terms of A and r
Using the formula for the n-th term of a GP, we can express the terms as follows:
- For the p-th term:
\[ 27 = A \cdot r^{p-1} \quad \text{(Equation 1)} \]
- For the q-th term:
\[ 8 = A \cdot r^{q-1} \quad \text{(Equation 2)} \]
- For the r-th term:
\[ 12 = A \cdot r^{r-1} \quad \text{(Equation 3)} \]

### Step 3: Form Ratios
Next, we can form ratios from these equations to eliminate \( A \):
1. Divide Equation 1 by Equation 3:
\[ \frac{27}{12} = \frac{A \cdot r^{p-1}}{A \cdot r^{r-1}} \implies \frac{27}{12} = r^{p-r} \implies \frac{9}{4} = r^{p-r} \quad \text{(Equation 4)} \]

2. Divide Equation 3 by Equation 2:
\[ \frac{12}{8} = \frac{A \cdot r^{r-1}}{A \cdot r^{q-1}} \implies \frac{12}{8} = r^{r-q} \implies \frac{3}{2} = r^{r-q} \quad \text{(Equation 5)} \]

### Step 4: Set Up Equations
Now, we have two equations:
- From Equation 4:
\[ r^{p-r} = \frac{9}{4} \quad \text{(1)} \]
- From Equation 5:
\[ r^{r-q} = \frac{3}{2} \quad \text{(2)} \]

### Step 5: Relate the Exponents
From these two equations, we can equate the powers of \( r \):
\[ r^{r-q} = r^{p-r} \implies r^{r-q} = r^{p-r} \]
This implies:
\[ r - q = p - r \implies p + q = 2r \quad \text{(Equation 6)} \]

### Step 6: Analyze the Solution
Since \( p, q, r \) are integers, Equation 6 suggests that for any integer \( r \), we can find corresponding integers \( p \) and \( q \). For example:
- If \( r = 2 \), then \( p + q = 4 \) can have pairs like (2, 2), (1, 3), etc.
- If \( r = 3 \), then \( p + q = 6 \) can have pairs like (3, 3), (2, 4), (1, 5), etc.

### Conclusion
Thus, there exist an infinite number of geometric progressions that can contain the terms 27, 8, and 12.