If a, b, c are in A.P. and x, y, z are in G.P., then prove that :
`x^(b-c).y^(c-a).z^(a-b)=1`

Explanation

a, b, c are in AP `rArr 2b=a+c`
x, y, z are in GP `rArr y^(2)=xz`.
`:. x^((b-c)).y^((c-a)).z^((a-b))`
`=x^((b-c)).(sqrt(xz))^((c-a)).z^(a-b)" "[ :' y=sqrt(xz)]`
`=x^((b-c)).x^(1/2(c-a)).z^(1/2(c-a)).z^((a-b))`
`=x^((b-c)+1/2(c-a)).z^(1/2(c-a)+(a-b))`
`=x^(1/2{2b-(a+c)}).z^(1/2{c+a-2b})=x^(0)xxz^(0)=1`.