The product of three numbers in G.P. is 216. If 2, 8, 6 be added to them, the results are in A.P. Find the numbers.

Let the three numbers be .

∴ According to the question

⇒ …(1)

⇒ a³ = 216

⇒ a = 6

2,8,6 is added to them

∴

The above sequence is in AP.

We know in AP.

2b = a + c

⇒

Substituting a = 6 in above equation we get,

⇒

⇒

⇒ 28r = 6 + 6r² + 8r

⇒ 6r² – 20r + 6 = 0

⇒ 6r² – 18r – 2r – 6 = 0

⇒ 6r(r – 3) – 2(r – 3) = 0

⇒ (6r – 2) (r – 3) = 0

⇒ r = 3 or r = 1/3 .

∴ Now the equation will be

⇒

⇒ 2,6,18 or 18,6,2

∴ The three numbers are 2,6,18.