If $x=\sum_{n=0}^{\infty} a^{n}, y=\sum_{n=0}^{\infty} b^{n}, z=\sum_{n=0}^{\infty} c^{n}$ where $a,

Question

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

Professor PC Thomas Classes · Answer

(a) : Given $|a|<1,|b|<1,|c|<1$, $a, b, c \in$ A.P.\ and $\sum_{n=0}^{\infty} a^{n}=\frac{1}{1-a}, \sum_{n=0}^{\infty} b^{n}=\frac{1}{1-b}$,\ $\sum_{n=0}^{\infty} c^{n}=\frac{1}{1-c}$\ $	herefore \quad x=\frac{1}{1-a}, \quad y=\frac{1}{1-b}, \quad z=\frac{1}{1-c}$\ $\Rightarrow a=\frac{x-1}{x}, b=\frac{y-1}{y}, c=\frac{z-1}{z}$\ as $a, b, c \in$ A.P. $	herefore 2 b=a+c$\ $2\left(\frac{y-1}{y}ight)=\frac{x-1}{x}+\frac{z-1}{z}$\ $\Rightarrow \frac{2}{y}=\frac{1}{x}+\frac{1}{z} \Rightarrow x, y, z \in$ H.P.\