Answer:
(x, y) = (0.5, -3) or (8, 12)
Explanation:
You want x and y such that 8, 2x, 2y is an arithmetic sequence and 2x, 2y, 36 is a geometric sequence.
Arithmetic sequence
The difference of successive terms is a constant for an arithmetic sequence.
2x -8 = 2y -2x
2x -4 = y . . . . . . . . divide by 2, add x
Geometric sequence
The ratio of successive terms is a constant for a geometric sequence.
2y/2x = 36/2y
y² = 18x . . . . . . . . . simplify and cross multiply
Solution
We can substitute the expression for y from the arithmetic sequence into the expression for y² in the geometric sequence:
(2x -4)² = 18x
4x² -16x +16 = 18x . . . . . expand
2x² -17x +8 = 0 . . . . . . . divide by 2, subtract 9x
(2x -1)(x -8) = 0 . . . . . . . factor
The values of x that make the factors zero are x = 1/2 and x = 8. The corresponding values of y are 2{1/2, 8} -4 = {1, 16} -4 = {-3, 12}.
The values of x and y are (1/2 and -3) or (8 and 12).
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Additional comment
The two sequences are ...
8, 1, -6 and 1, -6, 36 — difference of -7, ratio of -6
or
8, 16, 24 and 16, 24, 36 — difference of 8, ratio of 3/2