Solution to Problem #151

A key obervation is that

1 + r² + r⁴ = (1 + r + r²)(1 − r + r²)

Squaring x + y + z = 7, dividing it by x² + y² + z² = 19, and canceling common factors gives

(1 + r + r²)/(1 − r + r²) = 49/19
or
15r² − 34r + 15 = 0
(5r − 3)(3r − 5) = 0

Hence r = 5/3 or 3/5. In the first case, substituting in the first equation gives a = 9/7 and in the second case we have a = 25/7. This gives the geometric sequences 9/7, 15/7, 25/7 and 25/7, 15/7, 9/7 (which are the reverse of each other).