Let a,d be the first term and common difference of an AP and b,r be the first term and common ratio of a GP. Then,x=a+(m−1)d and x=brm−1y=a+(n−1)d and y=brn−1z=a+(p−1)d and z=brp−1​Now, x−y=(m−n)d,y−z=(n−p)d and z−x=(p−m)dAgain now, xy−z⋅yz−x⋅zx−y=[brm−1](n−p)d⋅[brn−1](p−m)d⋅[brp−1](m−n)d=b[n−p+p−m+m−n]d⋅r[(m−1)(n−p)+(n−1)(p−m)+(p−1)(m−n)]d=b0⋅r0=1​