Final answer:
To find the common ratio of the G.P., the relationship between terms in both A.P. and G.P. is established and solved for 'r'. After solving the equations, we find that 'r' could either be 1 or -2/3. However, since the numbers in the G.P. are distinct, 'r' cannot be 1, and -2/3 is not an option provided, indicating a possible error in the question or options.
Step-by-step explanation:
If x, 2y, 3z are in an arithmetic progression (A.P.), and the distinct numbers x, y, z are in a geometric progression (G.P.), we want to find the common ratio of the G.P.
For numbers to be in an arithmetic progression, the difference between consecutive terms must be constant. This implies:
For the numbers x, y, z to be in geometric progression with a common ratio r, it means:
From the G.P. we have:
Substituting these into the A.P. equation, we get:
- 2y - (y/r) = 3yr - 2y
- 2y - y/r = 3yr - 2y
- 2 + 1/r = 3r
- 3r² - r - 2 = 0
- (r - 1)(3r + 2) = 0
Therefore, r = 1 or r = -2/3. Since the numbers are distinct, r cannot be 1. Thus, the only feasible solution is the negative ratio which is not an option presented so there might be an error in the question or options provided.