Let the three numbers be .

∴ According to the question

⇒ …(1)

⇒ …(2)

From 2 we get

⇒ a³ = 1728

∴ a = 12.

From 1 we get

⇒

⇒ a + ar + ar² = 65r …(3)

Substituting a = 12 in 3 we get

⇒ 12 + 12r + 12r² = 38r

⇒ 12r² – 26r + 12 = 0…(4)

Dividing (4) by 2 we get

⇒ 6r² – 13r + 6 = 0

⇒ 6r² – 12r – r + 6 = 0

⇒ 6r(r – 2) – 1(r – 2) = 0

∴ r = 2 or r = 1/2

∴ Now the equation will be

⇒

⇒ 6,12,24 or 24,12,6.

∴ The three numbers are 6,12,24.