Question 439410
Let a, ar, and {{{ar^2}}} be the terms in geometric sequence.

Then from the given, {{{a*ar*ar^2 = a^3r^3 = (ar)^3 = -1000}}}, or ar = -10, after taking cube roots. 
==> a = -10/r.
Also, {{{a + ar + ar^2 = 15}}}<==> {{{a(1+r+r^2) = 15}}}
<==> {{{(-10/r)(1+r+r^2) = 15}}}
<==> {{{0 = r^2 + 5r + 2}}}, after cross-multiplying and simplifying.
<==> (2r+1)(r+1) = 0
==> r = -1/2 or -2
==> a = 20 or 5, respectively.
Hence the sequences are {20, -10, 5} and { 5, -10, 20}.  (The two sequences are DIFFERENT because they have different first terms and common ratios.)