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Question
If a, b, c, d are in continued proportion, prove that: (a2 – b2) (c2 – d2) = (b2 – c2)2
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Solution
a, b, c, d are in continued proportion
∴ `a/b = b/c = c/d` = k(say)
∴ c = dk, b = ck = dk. k = dk2,
a = bk = dk2. k = dk3
L.H.S. = (a2 – b2) (c2 – d2)
= [(dk3)2 – (dk2)2] [(dk)2 – d2]
= (d2k6 – d2k4) (d2k2 – d2)
= d2k4 (k2 – 1) d2(k2 – 1)
= d4k4 (k2 – 1)2
R.H.S. = (b2 – c2)2
= [(dk2)2 – (dk)2]2
= [d2k2 – d2k2]2
= [d2k2 (k2 – 1)]2
= d4k4(k2 – 1)2
∴ L.H.S. = R.H.S.
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