If a, b, c are in G.P. and 4a, 5b, 4c are in A.P. such that a + b + c

1.	If a, b, c are in G.P. and 4a, 5b, 4c are in A.P. such that a + b + c = 70, then what is the value of the smallest of the numbers a, b and c?
Answer» 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.

If a, b, c are in G.P. and 4a, 5b, 4c are in A.P. such that a + b + c = 70, then what is the value of the smallest of the numbers a, b and c?

Answer»

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.