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Question
If a, b, c, d are in continued proportion, prove that: a : d = triplicate ratio of (a – b) : (b – c)
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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
a : d = triplicate ratio of (a – b) : (b – C)
= (a – b)3 : (b – c)3
L.H.S. = a : d
= `a/d`
= `(dk^3)/d`
= k3
R.H.S. = `(a - b)^3/(b - c)^3`
= `(dk^3 - dk^2)^3/(dk^2 - dk)^3`
= `(d^3k^6(k - 1)^3)/(d^3k^3(k - 1)^3`
= k3
∴ L.H.S. = R.H.S.
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