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If A, B, C Are in Continued Proportion, Prove that (A^2 + B^2 + C^2)/(A + B + C)^2 = (A - B + C)/(A + B + C) - Mathematics

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Question

If a, b, c are in continued proportion, prove that `(a^2 + b^2 + c^2)/(a + b + c)^2  = (a - b + c)/(a + b + c)`

Solution

Given, a, b and c are in continued proportion.

`=> a/b = b/c = k` (say)

`=> a = bk, b = ck`

`=> a = (ck)k = ck^2, b = dk`

LHS = `(a^2 + b^2 + c^2)/(a + b + c)^2`

`= ((ck^2)^2 + (ck)^2 + c^2)/(ck^2 + ck + c)^2`

`= (c^2k^2 + c^2k^2 + c^2)/(c^2(k^2 + k + 1)^2)`

`= (c^2(k^4 + k^2 + 1))/(c^2(k^2 + k + 1))`

`= (k^2 + k^2 + 1)/(k^2 + k + 1)^2`

RHS  `(a - b + c)/(a + b + c)`

`= (ck^2 - ck + c )/(ck^2 + ck + c)`

`= (k^2 = k^2 + 1)/(k^2 + k + 1)^2`

LHS = RHS

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APPEARS IN

 Selina Solution for Concise Mathematics for Class 10 ICSE (2020 (Latest))
Chapter 7: Ratio and Proportion (Including Properties and Uses)
Exercise 7(C) | Q: 10.2 | Page no. 101

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Solution If A, B, C Are in Continued Proportion, Prove that (A^2 + B^2 + C^2)/(A + B + C)^2 = (A - B + C)/(A + B + C) Concept: Proportions.
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