Let the three numbers be a/r, a, ar
So, according to the question
a/r + a + ar = 13/12 … equation (1)
a/r × a × ar = -1 … equation (2)
From equation (2) we get,
a3 = -1
a = -1
From equation (1) we get,
(a + ar + ar2)/r = 13/12
12a + 12ar + 12ar2 = 13r … equation (3)
Substituting a = – 1 in equation (3) we get
12( – 1) + 12( – 1)r + 12( – 1)r2 = 13r
12r2 + 25r + 12 = 0
12r2 + 16r + 9r + 12 = 0… equation (4)
4r (3r + 4) + 3(3r + 4) = 0
r = -3/4 or r = -4/3
Now the equation will be
-1/(-3/4), -1, -1×-3/4 or -1/(-4/3), -1, -1×-4/3
4/3, -1, 3/4 or 3/4, -1, 4/3
∴ The three numbers are 4/3, -1, 3/4 or 3/4, -1, 4/3