Question

If p,q,r are in A.P. and x,y,z are in G.P. ,prove that xq-ryr-pzp-q=1.

Answer 1

p,q,r in AP means, 2q=p+r....(i)
and q--p= r--q = common difference....(ii)
x,y,z in GP means y^2=xz....(iii)

Now, as per question, x^(q-r). y^(r-p). z^(p-q)
= x^(p-q). y^(r-p). z^(p-q) .....using(ii)
=(xz)^(p-q) . y^(r-p)
= y^2(p-q) . y^(r-p) ....using(iii)
= y^(2p-2q+r-p)
=y^(p+r-2q)
=y^(2q-2q) ....using(i)
=y^(0)
=1

Answer 2

p,q,r are in AP  so 2q = p+r

 and x,y ,z are in gp so y^2 = 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)

= x^0 z^0

=1