if a, b, c, d, e are in continued proportion, prove that a^2+b^2+c^2+d^2,ab+bc+cd+de,b^2+c^2+d^2+e^2
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(i) Since a, b, c, d are in continued proportion then
a/b = b/c = c/d = k
⇒ a = bk, b = ck , c = dk
⇒ a = ck2
⇒ a = dk3, b = dk2 and c = dk
(ii) 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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