If a, b, c, d are in continued proportion, prove that :
(a² − b²)(c² − d²) = (b² − c²)²

Question

If a, b, c, d are in continued proportion, prove that : (a^2 − b^2)(c^2 − d^2) = (b^2 − c^2)^2

KnowledgeBoat · Accepted Answer

Given a, b, c, d are in continued proportion. ∴ a : b = b : c = c : d = k (let) ⇒ c = dk, b = ck = (dk)k = dk2, a = bk = (dk2)k = dk3. Substituting values of a, b and c in L.H.S. of (a2 − b2)(c2 − d2) = (b2 − c2)2, we get : ⇒ (a2 - b2)(c2 - d2) ⇒ [(dk3)2 - (dk2)2] [(dk)2 - d2] ⇒ [d2k6 - d2k4] [d2k2 - d2] ⇒ d4(k6 - k4)(k2 - 1) ⇒ d4 k4(k2 - 1)(k2 - 1) ⇒ d4 k4 (k2 - 1)2. Substituting values of a, b and c in R.H.S. of (a2 − b2)(c2 − d2) = (b2 − c2)2, we get : ⇒ (b2 - c2)2 ⇒ [(dk2)2 - (dk)2]2 ⇒ [d2k4 - d2k2]2 ⇒ [d2k2(k2 - 1)]2 ⇒ d4 k4 (k2 - 1)2. Since, L.H.S. = R.H.S. Hence, proved that (a2 − b2)(c2 − d2) = (b2 − c2)2 .

Mathematics

If a, b, c, d are in continued proportion, prove that :
(a² − b²)(c² − d²) = (b² − c²)²

Ratio Proportion

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Mathematics

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