Solution to the Problem:
The answer is 18, 27, 36 and 48.
Since the four terms are distinct, and the first three terms are in
arithmetic sequence, we must have the first three terms as y-b, y and y+b. Since all the four terms are positive, it follows that y > b.
Now, the last three terms are in geometric sequence. Thus, the common ratio is (y+b)/y, so that the 4th term is (y+b)^2/y.
Now, the difference between the last term and the first is 30, and so:
(y+b)^2/y - (y-b) = 30
or, b(3 + b/y) = 30
Since b< y, it follows that:
30 = b(3 + b/y) < 4*b, and:
30 = b(3 + b/y) > 3*b, so that:
7.5< b< 10.
The only values that satisfy the above inequality occur at b = 8, 9
Substituting b = 8 in (*), we have:
24 + 64/y = 30, or y = 32/3, which is not an integer and thus leads to a contradiction.
Substituting b = 9 in (*), we have:
24 + 81/y = 30, or y = 27, so that;
(y+b)^2/y = 36^2/27 = 48, and (y-b, y, y+b) = (18, 27, 36)
Thus, the required four terms of the given sequence are
18, 27, 36 and 48.
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