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. $HP$
• B. $Arithmetico-GeometricProgression$
• C. $AP$
• D. $GP$

Answer 1

The correct option is A $HP$

Given that,

$x=\sum _{n=0}^{\infty }{a}^n,y=\sum _{n=0}^{\infty },z=\sum _{n=0}^{\infty }{c}^n$

Now,

$\begin{array}{c}x=1+a+a^2+.............\infty \\ y=1+b+b^2+.............\infty \\ z=1+c+c^2+.............\infty \\ x=\frac{1}{1-a}\\ y=\frac{1}{1-b}\\ z=\frac{1}{1-c}\end{array}$

Given $a,b,c$ are in $A.P$

$1-a,1-b,1-c$ are in $A.P$

$\frac{1}{1-a},\frac{1}{1-b},\frac{1}{1-c}$ are in $H.P$

Option $A$ is correct answer.