Question

if a, b, c, d are continued proportion, prove that:

(i) (a + b)(b + c) - (a + c) (b + d) = (b - c)^2
(ii) (a^2 - b^2) (c^2 - d^2) = (b^2 - c^2)^2
a=dk^3, b=dk^2, c= dk

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If a, b, c, d are in continued proportion, prove that (b – c)2 + (c – a)2 + (d – b)2 = (d – a)2

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asked Feb 23, 2019 in Class X Maths by navnit40 (-4,938 points)

If a, b, c, d are in continued proportion, prove that (b – c)2 + (c – a)2 + (d – b)2 = (d – a)2

ratio and proportion

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answered Feb 23, 2019 by aditya23 (-2,137 points)

Since a, b, c, d are in continued proportion, we have

a/b = b/c = c/d = K(say)

∴ c = dK, b = cK = dK2 and a = bK = dK3

L.H.S. = (b – c)2 + (c – a)2 + (d – b)2

= (dK2 – dK)2 + (dK – dK3)2 + (d – dK2)2

= d2K2(K – 1)2 + d2K2(1 – K2)2 + d2(1 – K2)2

= d2[K2(K – 1)2 + K2(K2 – 1)2 + d2(K2 – 1)2]

= d2[K2(K – 1)2 + K2(K – 1)2(K + 1)2 + (K – 1)2(K + 1)2]

= d2(K – 1)2 [K2 + K2 (K + 1)2 + (K + 1)2]

= d2(K – 1)2[K2 + K2(K2 + 2K + 1) + K2 + 2k + 1]

= d2(K – 1)2[K4 + 2K3 + 3K2 + 2K + 1]

= d2(K – 1)2 (K2 + K + 1)2

= d2[(K – 1)(K2 + K + 1)]2

= d2(K3 – 1)2 = (dK3 – d)2 = (a – d)2 = (d – a)2 = R.H.S.

Hence, (b – c)2 + (c – a)2 + (d – b)2 = (d – a)2

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