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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

The sum of the numbers \(1+2.2+3.2^{2}+4.2^{3}+\ldots+50.2^{49}\) is (A) \(1+49.2^{49}\) (B) \(1+49.2^{50}\) (C) \(1+50.2^{49}\) (D) \(1+50.2^{50}\)

If \(\mathrm{a}_{1}, \mathrm{a}_{2}, \ldots \mathrm{a}_{10}\) be in A. P., \(\left(1 / \mathrm{h}_{1}\right),\left(1 / \mathrm{h}_{2}\right) \ldots\left[1 / \mathrm{h}_{10}\right)\) be in A. \(P\) and \(a_{1}=h_{1}=2, a_{10}=h_{10}=3\) then \(a_{4} h_{7}=\) (A) \((1 / 6)\) (B) 6 (C) 3 (D) 2

If the function \(\mathrm{f}\) satisfies the relation \(\mathrm{f}(\mathrm{x}+\mathrm{y})=\mathrm{f}(\mathrm{x}) \mathrm{f}(\mathrm{y})\) for all \(\mathrm{x}, \mathrm{y} \in \mathrm{N}\), Further if \(\mathrm{f}(1)=3\) and \(n_{r=1} f(a+r)=(81 / 2)\left(3^{n}-1\right)\) then \(a=\) (A) 4 (B) 2 (C) 1 (D) 3

If an A. P., \(\mathrm{T}_{35}=-50\) and \(\mathrm{d}=-3\) then \(\mathrm{S}_{35}=\) (A) 35 (B) 38 (C) 32 (D) 29

\(1+5+14+30+\ldots \mathrm{n}\) terms \(=\) (A) \([\\{(n+2)(n+3)\\} /(12)]\) (B) \([\\{n(n+1)(n+5)\\} /(12)]\) (C) \([\\{n(n+2)(n+3)\\} /(12)]\) (D) \(\left[\left\\{n(n+1)^{2}(n+2)\right\\} /(12)\right]\)
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