Mathematics - Sequence & Series Question with Solution | TestHub

$$\begin{array}{ll}x={\sum }_{n=0}^{\infty }{a}^n=\frac{1}{1-a}& \Rightarrow a=1-\frac{1}{x}\\ y={\sum }_{n=0}^{\infty }{b}^n=\frac{1}{1-b}& \Rightarrow b=1-\frac{1}{y}\end{array}$$ $$\begin{array}{rl}& z=\sum _{n=0}^{\infty }{c}^n=\frac{1}{1-c}\Rightarrow c=1-\frac{1}{z}\\ & a,b,c\text{ are in A.P. }\\ & 2b=a+c\\ & 2\left(1-\frac{1}{y}\right)=1-\frac{1}{x}+1-\frac{1}{y}\\ & \frac{2}{y}=\frac{1}{x}+\frac{1}{z}\\ & \Rightarrow x,y,z\text{ are in H.P. }\end{array}$$