Let the three numbers be a/r, a, ar
So, according to the question
a/r + a + ar = 65 … equation (1)
a/r × a × ar = 3375 … equation (2)
From equation (2) we get,
a3 = 3375
a = 15.
From equation (1) we get,
(a + ar + ar2)/r = 65
a + ar + ar2 = 65r … equation (3)
Substituting a = 15 in equation (3) we get
15 + 15r + 15r2 = 65r
15r2 – 50r + 15 = 0… equation (4)
Dividing equation (4) by 5 we get
3r2 – 10r + 3 = 0
3r2 – 9r – r + 3 = 0
3r(r – 3) – 1(r – 3) = 0
r = 3 or r = 1/3
Now, the equation will be
15/3, 15, 15×3 or
15/(1/3), 15, 15×1/3
So the terms are 5, 15, 45 or 45, 15, 5
∴ The three numbers are 5, 15, 45.