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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
As a, b, c, are in continued proportion
Let `a/b = b/c` = k
L.H.S. = (a + b + c) (a – b + c)
= (ck2 + ck + c)(ck2 – ck + c)
= c(k2 + k + 1)c(k2 – k + 1)
= c2(k2 + k + 1)(k2 – k + 1)
= c2(k4 + k2 + 1)
R.H.S. = a2 + b2 + c2
= (ck2)2 + (ck)2 + (c)2
= c2k4 + c2k2 + c2
= c2(k4 + k2 + 1)
∴ L.H.S. = R.H.S.
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