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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.
Solution
(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.
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