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

$\begin{array}{l}\mathrm{x}=1+\mathrm{a}+{\mathrm{a}}^2=\dots \dots \dots .\\ \\ x=\frac{1}{1-a}\Rightarrow a=1-\frac{1}{x}\\ \\ y=\frac{1}{1-b}\Rightarrow b=1-\frac{1}{y}\\ \\ z=\frac{1}{1-c}\Rightarrow c=1-\frac{1}{z}\\ \\ \mathrm{a},\mathrm{b},\mathrm{c} are in A.P.\\ \\ \Rightarrow 1-\frac{1}{x},1-\frac{1}{y},1-\frac{1}{z} are in A.P.\\ \\ \Rightarrow -\frac{1}{x},-\frac{1}{y},-\frac{1}{z} are in A.P.\\ \\ \Rightarrow \frac{1}{x},\frac{1}{y},\frac{1}{z} are in A.P.\end{array}$