Question

If a, b, c are in A.P. and x, y, z are in G.P., then the value of x^{b − c} y^{c − a} z^{a − b} is
(a) 0
(b) 1
(c) xyz
(d) x^a y^b z^c

Answer 1

(b) 1

$$\begin{gathered} a,b\mathrm{and}c\mathrm{are}\mathrm{in}\mathrm{A}.\mathrm{P}. \\ \therefore 2b=a+c........\left(\mathrm{i}\right) \\ \mathrm{And},x,y\mathrm{and}z\mathrm{are}\mathrm{in}\mathrm{G}.\mathrm{P}. \\ \therefore y^2=xz \\ \mathrm{Now},x^{b-c}{y}^{c-a}{z}^{a-b} \\ =x^{b+a-2b}{y}^{2b-a-a}{z}^{a-b}\left[\mathrm{From}\left(\mathrm{i}\right)\right] \\ =x^{a-b}{y}^{2\left(b-a\right)}{z}^{a-b} \\ ={\left(xz\right)}^{a-b}{\left(xz\right)}^{b-a}\left[\mathrm{From}\left(\mathrm{ii}\right),y^2=xz\right] \\ ={\left(\mathrm{xz}\right)}^0 \\ =1 \end{gathered}$$