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Question
If a, b, c, d are in continued proportion, prove that:
(a2 + b2 + c2) (b2 + c2 + d2) = (ab + bc + cd)2.
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Solution
L.H.S.
= (d2k6 + d2k4 + d2k2)(d2k4 + d2k2 + d2)
= d2k2 (k4 + k2 + 1)d2 (k4 + k2 + 1)
= d4k2 (k4 + k2 + 1)2
R.H.S.
= (ab + bc + cd)2
= (dk3·dk2 + dk2·dk + dk·d)2
= d4·k2 (k4 + k2 + 1)2
L.H.S. = R.H.S.
Hence proved.
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