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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?

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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.

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