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Question

If x,yandz in AP, then 1x+y,1z+x,1y+z are in:


A

AP

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B

GP

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C

HP

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D

AP and HP

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Solution

The correct option is A

AP


Explanation for the correct option:

Step 1: Express as per the condition of AP:

It is given that x,yandz in AP, so

y-x=z-y.....1

By taking reciprocal of 1, we get

1y-x=1z-y1x-y=1y-z.......(2)

Step 2: Use the algebraic identity a2-b2=(a+b)(a-b):

We can write 2 as

1x+yx-y=1y+zy-zy-zx+y=x-yy+z

Add and subtract x on L.H.S and y on R.H.S:

x+y-z-xx+y=z+x-y-zy+zx+y-z+xx+y=z+x-y+zy+z

Divide both side by z+x:

x+y-z+xx+yz+x=z+x-y+zy+zz+x1z+x-1x+y=1y+z-1z+x

Therefore, 1x+y,1z+x,1y+z are in AP.

Hence, option A is correct.


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