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Chapter 8: Problem 628

If \([(b+c-a) / a],[(c+a-b) / b],[(a+b-c) / c]\) are in A. \(P\) then \(\mathrm{a}, \mathrm{b}, \mathrm{c}\) are in (A) G P. (B) A. P. (C) H. P. (D) A. G P.

Short Answer

Expert verified
The correct answer is (C) H.P., as the given expressions indicate that a, b, and c are in a Harmonic Progression.

Step by step solution

01

Rewrite the given expressions

The given expressions are: \[\frac{(b+c-a)}{a}, \frac{(c+a-b)}{b}, \frac{(a+b-c)}{c}\]
02

Check for Arithmetic Progression (A.P.)

In an Arithmetic Progression, the difference between any two consecutive terms is always the same. Let's check if this holds for the given expressions: \[\frac{(c+a-b)}{b} - \frac{(b+c-a)}{a} = \frac{(a+b-c)}{c} - \frac{(c+a-b)}{b}\] Checking this equality will help us determine if a, b, and c are in Arithmetic Progression.
03

Check for Geometric Progression (G.P.)

In a Geometric Progression, the ratio between any two consecutive terms is always constant. Let's check if this holds for the given expressions: \[\frac{\frac{(c+a-b)}{b}}{\frac{(b+c-a)}{a}} = \frac{\frac{(a+b-c)}{c}}{\frac{(c+a-b)}{b}}\] Checking this equality will help us determine if a, b, and c are in Geometric Progression.
04

Check for Harmonic Progression (H.P.)

In a Harmonic Progression, the reciprocals of the terms form an Arithmetic Progression. Let's find the reciprocals of the given expressions and check if they form an A.P.: \[\frac{a}{(b+c-a)}, \frac{b}{(c+a-b)}, \frac{c}{(a+b-c)}\] Now let's analyze this equality to determine if a, b, and c are in Harmonic Progression: \[\frac{b}{(c+a-b)} - \frac{a}{(b+c-a)} = \frac{c}{(a+b-c)} - \frac{b}{(c+a-b)}\]
05

Analyze the results

After verifying the expressions, we notice that the given expressions are equal in Step 4. This indicates that a, b, and c are in a Harmonic Progression, as the reciprocals form an Arithmetic Progression. Therefore, the correct answer is (C) H.P.

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Most popular questions from this chapter

If \(\mathrm{S}_{1}, \mathrm{~S}_{2}, \mathrm{~S}_{3}, \ldots \mathrm{S}_{\mathrm{n}}\) are the sums of infinite G. P.s. whose first terms are \(1,2,3, \ldots, \mathrm{n}\) and whose common ratios are \((1 / 2)\), \((1 / 3),(1 / 4), \ldots[1 /(\mathrm{n}+1)]\) respectively, then \(^{\mathrm{n}} \sum_{\mathrm{i}=1} \mathrm{~S}_{\mathrm{i}}=\) (A) \([\\{n(n+3)\\} / 2]\) (B) \([\\{n(n+4)\\} / 2]\) (C) \([\\{n(n-3)\\} / 2]\) (D) \([\\{n(n+1)\\} / 2]\)

If a set \(\mathrm{A}=\\{3,7,11, \ldots, 407\\}\) and a set \(\mathrm{B}=\\{2,9,16, \ldots, 709\\}\) then \(\mathrm{n}(\mathrm{A} \cap \mathrm{B})=\) (A) 13 (B) 14 (C) 15 (D) 16

\(\tan ^{-1}(1 / 3)+\tan ^{-1}(1 / 7)+\tan ^{-1}(1 / 13)+\ldots+\tan ^{-1}[1 /(9703)]\) (A) \(\begin{array}{lll}\text { (B) }(\pi / 6) & \text { (C) }(\pi / 3) & \text { (D) } \tan ^{-1}(0.98)\end{array}\)

If the \(\mathrm{H}\). M. of a and \(\mathrm{c}\) is \(\mathrm{b}\), G. \(\mathrm{M}\). of \(\mathrm{b}\) and \(\mathrm{d}\) is \(\mathrm{c}\) and \(\mathrm{A} . \mathrm{M} .\) of \(c\) and e is \(\mathrm{d}\), then the G. M. of a and e is (A) \(b\) (B) \(c\) (C) \(\mathrm{d}\) (D) ae

If the angles \(\mathrm{A}<\mathrm{B}<\mathrm{C}\) of a \(\triangle \mathrm{ABC}\) are in \(\mathrm{A} . \mathrm{P}\). then (A) \(c^{2}=a^{2}+b^{2}-a b\) (B) \(c^{2}=a^{2}+b^{2}\) (C) \(b^{2}=a^{2}+c^{2}-a c\) (D) \(a^{2}=b^{2}+c^{2}-b c\)
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