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 if there exists a geometric progression (GP) containing the terms 27, 8, and 12, and to find how many such progressions are possible, we can follow these steps:

### Step 1: Assume a First Term
Let's assume that 8 is the first term of the GP. We denote this as:
- First term (A) = 8

### Step 2: Define the Terms in the GP
Let 12 and 27 be the m-th and n-th terms of the GP, respectively. The general formula for the n-th term of a GP is given by:
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