(a) When the sequence is arithmetic, sequential terms have a common difference.
... x - 3 = (4x/3) - x . . . . differences of sequential terms are equal
... (2/3)x = 3 . . . . . . . add 3-(1/3)x
... x = 9/2 . . . . . . . . . multiply by 3/2
(The arithmetic sequence is 3, 4.5, 6. The common difference is 3/2.)
(b) When the sequence is geometric, sequential terms have a common ratio.
... x/3 = (4x/3)/x . . . . . ratios of sequential terms are equal
... x^2 = 4x . . . . . multiply by 3x
... x = 4 . . . . . . . . divide by x. (the "solution" x=0 is extraneous)
(The geometric sequence is 3, 4, 16/3. The common ratio is 4/3.)