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Question

If a,b,c be in G.P and a+x,b+x,c+x, in H.P , then the value of x is (a,b,c are distinct numbers)


A

c

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B

b

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C

a

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D

noneofthese

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Solution

The correct option is B

b


Step 1: Apply the G.P relationship on a,b,c.

We have givena,b,c be in G.P anda+x,b+x,c+x, in H.P

We need to find the value of x.

asa,b,c be in G.P

āˆ“b2=ac....(1)

Step 2: Apply H.P relationship on a+x,b+x,c+x.

asa+x,b+x,c+x, in H.P.

āˆ“1a+x,1b+xand1c+xareinA.PSo,1b+x-1a+x=1c+x-1b+x(a+x)-(b+x)(a+x)(b+x)=(b+x)-(c+x)(c+x)(b+x)

Step 3: Simplifying the above equation.

ā‡’(a-b)(a+x)=(b-c)(c+x)bycancelling(b+x)frombothsides.ā‡’(a-b)(c+x)=(a+x)(b-c)ā‡’ac+ax-bc-bx=ab-ac+bx-cxā‡’2ac-bc-ab+ax-2bx+cx=0ā‡’2b2-bc-ab+(a-2b+c)x=0āˆµb2=acā‡’b(2b-c-a)=(2b-a-c)x(2b-c-a)canceloutbothsidesā‡’b=x

Hence Option (B)is the correct option.


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