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Question
If a, b, c are in continued proportion, prove that (a + b + c) (a – b + c) = a2 + b2 + c2
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Solution
Given that a, b and c are in continued proportion
`=> a/b = b/c => b^2 = ac`
L.H.S = (a + b + c)(a - b + c)
= a(a - b + c) + b(a - b + c) + c(a - b + c)
`= a^2 - ab + ac + ab - b^2 + bc + ac - bc + c^2`
`= a^2 + ac - b^2 + ac + c^2`
`= a^2 + b^2 - b^2 + b^2 + c^2` [∵ `b^2 = ac`]
`= a^2 + b^2 + c^2``
=R.H.S
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