If $\mathrm{x}=\sum _{\mathrm{n}=0}^{\infty }{\mathrm{a}}^{\mathrm{n}}, \mathrm{y}=\sum _{\mathrm{n}=0}^{\infty }{\mathrm{b}}^{\mathrm{n}}, \mathrm{z}=\sum _{\mathrm{n}=0}^{\infty }{\mathrm{c}}^{\mathrm{n}}$ where a, b, c are in A.P. and $\left|\mathrm{a}\right|<1, \left|\mathrm{b}\right|<1, \left|\mathrm{c}\right|<1$ then x, y, z are in

$$\begin{gathered} \mathrm{x}=1+\mathrm{a}+{\mathrm{a}}^2.....\infty =\frac{1}{1-\mathrm{a}} \\ \mathrm{y}=\frac{1}{1-\mathrm{b}} \mathrm{z}=\frac{1}{1-\mathrm{c}} \\ \begin{array}{c}\mathrm{a}, \mathrm{b}, \mathrm{c} \to \mathrm{A}.\mathrm{P}\\ -\mathrm{a}, -\mathrm{b}, -\mathrm{c} \to \mathrm{A}.\mathrm{P}\\ 1-\mathrm{a}, 1-\mathrm{b}, 1-\mathrm{c} \to \mathrm{A}.\mathrm{P}\end{array} \overline{) \begin{array}{c}\frac{1}{1-\mathrm{a}}, \frac{1}{1-\mathrm{b}}, \frac{1}{1-\mathrm{c}} \to \mathrm{H}.\mathrm{P}\\ \mathrm{x}, \mathrm{y}, \mathrm{z} \to \mathrm{H}.\mathrm{P}\end{array}} \end{gathered}$$