All the terms of an increasing geometric progression are positive integers and no two of them are equal. If product of its first five terms is P and P = (12)5, then the common ratio of this G.P. is :

The problem is asking for the common ratio of a geometric progression (G.P.) where the product of the first five terms is equal to (12)^5.

Step 1: Understand the problem In a geometric progression, each term after the first is found by multiplying the previous term by a fixed, non-zero number called the common ratio. If we denote the first term as 'a' and the common ratio as 'r', then the terms of the G.P. are a, ar, ar^2, ar^3, ar^4, and so on.

Step 2: Set up the equation The problem states that the product of the first five terms is (12)^5. So, we can set up the equation as follows: a * ar * ar^2 * ar^3 * ar^4 = (12)^5

Step 3: Simplify the equation This simplifies to: a^5 * r^10 = (12)^5

Step 4: Solve for 'r' Since all terms of the G.P. are positive integers and no two of them are equal, 'a' must be 1 (because 'a' is the smallest term and it's a positive integer). So, we can substitute 'a' with 1 in the equation: 1^5 * r^10 = (12)^5 This simplifies to: r^10 = (12)^5

Taking the fifth root of both sides gives us: r^2 = 12

Finally, taking the square root of both sides gives us: r = sqrt(12)

So, the common ratio of the G.P. is sqrt(12).