Question

If (a + 2 b + c), (a – c) and (a – 2 b + c) are in continued proportion, prove that b is the mean proportional between a and c.

Answer 1

(a + 2 b + c), (a – c) and (a – 2 b + c) are in continued proportion
⇒ `(a + 2b + c)/(a - c) = (a - c)/(a - 2b + c)`

∴ `(a + 2b + c)/(a - c) = (a - c)/(a - 2b + c)`

⇒ (a + 2b + c)(a - 2b + c) = (a - c)²
⇒ a² – 2ab + ac + 2ab – 4b² + 2bc + ac – 2bc + c² = a² – 2ac + c²
⇒ a² – 2ab + ac + 2ab – 4b² + 2bc + ac – 2bc + c² – a² + 2ac – c² = 0
⇒ 4ac – 4b² = 0
⇒ ac – b² = 0
⇒ b² = ac
Hence b is the mean proportional between a and c.