Suppose are not consecutive terms of an arithmetic or geometric progression.
For the arithmetic progression:
Let (1)
(2)
(3)
Substracting (2) from (1) and (3) from (2) we get
Then
But, the left side member is irational and the right side member is rational.
For the geometric progression:
Let (1)
(2)
(3)
Then
The last equality is not true.
It is just that you were doing a different problem than Red_dog. You were saying that sqrt(2),sqrt(3),sqrt(5) cannot be right next to each other, and what you did is correct. But Red_dog did a stronger problem he showed that you cannot have these three numbers in any arithmetic progession. Meaning you cannot have sqrt(2) as a 3rd term, sqrt(3) as a 10th term, and sqrt(5) as a 29th term.