If p,q,r are in A.P. and x,y,z are in G.P. ,prove that x^q-ry^r-pz^p-q=1.

p,q,r are in AP  so 2q = p+r

 and x,y ,z are in gp so y² = xz

  now x^q-r y^r-p   z^p-q = x^{(p+r / 2  - r )} y^r-p   z^{(p - (p- r)/2 )}

  =x^{(p- r/2)}. y ^r-p z^(p-r/2)

=  x^{(p- r/2)}. (xz) ^(r-p)/2 z^(p-r/2)

= x^{(p- r/2)}. x ^(r-p/2) z^(r-p/2) z^(p-r/2)

= x⁰ z⁰

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