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Question
If a, b, c, d are in continued proportion, prove that: `(a^3 + b^3 + c^3)/(b^3 + c^3 + d^3) = a/d`
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. = `(a^3 + b^3 + c^3)/(b^3 + c^3 + d^3)`
= `((dk^3)^3 + (dk^2)^3 + (dk)^3)/((dk^2)^3 + (dk)^3 + d^3)`
= `(d^3k^9 + d^3k^6 + d^3k^3)/(d^3k^6 + d^3k^3 + d^3)`
= `(d^3k^3(k^6 + k^3 + 1))/(d^3(k^6 + k^3 + 1)`
= k3
R.H.S. = `a/d`
= `(dk^3)/d`
= k3
∴ L.H.S. = R.H.S.
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