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 AP
and |a| <1,|b|<1,|c|<1 then x,y,z are in

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

Answer 1

The correct option is D H.P

$x={\sum }_{n=0}^{\infty }{a}^n=\frac{1}{1-a}ora=1-\frac{1}{x}y={\sum }_{n=0}^{\infty }{b}^n=\frac{1}{1-b}orb=1-\frac{1}{y}z={\sum }_{n=0}^{\infty }{c}^n=\frac{1}{1-c}orc=1-\frac{1}{z}$
a,b,c are in A.P.
$\to 2b=a+c$
$2(1-\frac{1}{y})=1-\frac{1}{x}+1-\frac{1}{y}\Rightarrow \frac{2}{y}=\frac{1}{x}+\frac{1}{z}\Rightarrow x,y,zareinH.P.$