a, b, c are in G.P. ⇒ b2 = ac ...(i)
4a, 5b, 4c are in A.P. ⇒ 2 x 5b = 4a + 4c
⇒ 10b = 4a + 4c ⇒ 5b = 2a + 2c ...(ii)
Also, given a + b + c = 70 ...(iii)
⇒ 2a + 2b + 2c = 140 ⇒ 5b + 2b = 140 (From (ii))
⇒ 7b = 140 ⇒ b = 20.
Now, from (i), 400 = ac. (∵ b = 20)
Also, from (ii), a + 20 + c = 70 ⇒ a + c = 50
∴ (a – c)2 = (a + c)2 – 4ac
= 2500 – 1600 = 900
⇒ a – c = ± 30
∴ a + c = 50 and a - c = ± 30 ⇒ a = 40, c = 10 or a = 10, c = 40.
∴ The least value out of a, b and c is 10.