If 1/x,1/y,1/z are AP, show that:

(i) yz,zx,xy are in AP

(ii) xy,zx,yz are in AP

(iii) y+z/x,z+x/y,x+y/z are in AP

I got the first one but how to do the other two?

given: 1/x , 1/y and 1/z are in AP.
multiplying each term with xyz will also produce an AP
xyzx ,xyzy,xyzz  are also in APyz , xz , xy are in AP
since reverse of AP is also in AP.
therefore
xy , xz , yz are also in AP

multiplying each term of given AP by (x+y+z)
‚Äčtherefore
x+y+zx ,x+y+zy , x+y+zz are in APx+y+zx-1 ,x+y+zy-1 , x+y+zz-1 are in AP  [subtracting 1 from each termy+zx ,x+zy,x+yz  are in AP

hope this helps you
 

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