Question

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

• A. H.P
• B. Arithmetic-Geometric progression
• C. A.P
• D. G.P

Answer 1

The correct option is A H.P

Given $\left|a\right|<1,\left|b\right|<1,\left|c\right|<1,a,b,c\in A.P$

and $\sum _{n=0}^{\infty }{a}^n=\frac{1}{1-a},\sum _{n=0}^{\infty }{b}^n=\frac{1}{1-b},\sum _{r=0}^{\infty }{c}^n=\frac{1}{1-c}$

$\therefore x=\frac{1}{1-a},y=\frac{1}{1-b},c=\frac{1}{1-c}$

$\Rightarrow a=\frac{x-1}{x},b=\frac{y-1}{y},c=\frac{z-1}{z}$

$\because 2b=a+casa,b,c\in A.P$

$2\left(\frac{y-1}{y}\right)=\frac{x-1}{x}+\frac{z-1}{z}\Rightarrow \frac{2}{y}=\frac{1}{x}+\frac{1}{z}$

$\Rightarrow x,y,z\in H.P$