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Question

Ifa,b&c are in Arithmetic Progression and a2,b2,c2are in Harmonic Progression, then


A

abc

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B

a2=b2=c22

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C

a,b&c are in G,P

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D

-a2,b,c are in G.P

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Solution

The correct option is D

-a2,b,c are in G.P


Explanation for correct option:

Step-1: Expressing the given data:

Given that,

a,b,c are in Arithmetic Progression

2b=a+c(i)

a2,b2,c2 are in harmonic progression

So, 1a2,1b2,1c2are in arithmetic progression

Step2. Finding progression.

1b21a2=1c21b2

(a2-b2)a2b2=(b2-c2)b2c2

(a+b)(a-b)a2=(b-c)(b+c)c2 a-b=b-c

(a+b)a2=(b+c)c2

a2(b+c)=c2(a+b)

a2b+a2c=c2a+c2b

ac(c-a)+b(c-a)(c+a)=0

(c-a)(ab+bc+ca)=0

(c-a)=0or(ab+bc+ca)=0

c=-aor(a+c)b+ca=0

Substitute (i) in (a+c)b+ca=0

2b2+ca=0

b2=-ac2

-a2,b,care in Geometric Progression

Hence, correct option is (D).


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