If `(a-x)/(p x)=(a-y)/(q y)=(a-z)/ra n dp ,q ,a n dr` are in A.P., then prove that `x ,y ,z` are in H.P.

Explanation

We have,
`(a-x)/(pq)=(a-y)/(qy)=(a-z)/(rz)=lamda` (say)
`rArrp=(a-x)/(lamdax),q=(a-y)/(lamday),r=(a-z)/(lamdaz)`
Now,p,q,r are in A.P. Therefore,
`(a-x)/(lamdax),(a-y)/(lamday),(a-z)/(lamdaz)` are in A.P.
`rArr(a-x)/x,,(a-y)/y,(a-z)/z` are in A.P.
`rArra/x-1,a/y-1,a/z-1` are in A.P.
`rArra/x,a/y,a/z` are in A.P.
`rArr1/x,1/y1/z` are in A.P.
`rArrx,y,z` are in H.P.