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If x ,y ,a n dz are pth, qth, and rth te...

If `x ,y ,a n dz` are pth, qth, and rth terms, respectively, of an A.P. nd also of a G.P., then `x^(y-z)y^(z-x)z^(x-y)` is equal to `x y z` b. 0 c. 1 d. none of these

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Given that x,y,z are the pth,qth and rth terms of an A.P. respectively.
`therefore` x=A+(p-1)D,
y=A+(q-1)D,
and z-x=(r-p)D
Also x,y,z are the pth,qth and rth terms of a GP.
Let a be the first term and R be the common ratio
`thereforex=aR^(p-1),`
`y=aR^(q-1),`
and `z=aR^(r-1)`
Now, `x^(y-z)y^(z-x)z^(x-y)=(aR^(q-1))^(z-x)(aR^(r-1))^(x-y)`
= `a^(y-z+z-x+x-y)R^((p-1)(y-z)+(q-1)(z-x)+(r-1)(x-y)`
`=a^(0)R^((r-1)(q-r)D+(q-1)(r-p)D+(r-1)(p-q)D)`
`=a^(0)R^(0)=1`
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