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 $\left|a\right|<1,\left|b\right|<1,\left|c\right|<1$

• A. GP
• B. AP
• C. Arithmetic-Geometric Progression
• D. HP

Answer 1

The correct option is D HP

Consider the given expression.

$x={\sum }_{n=0}^{\infty }{a}^n$

$x=1+a+a^2+a^3+.....+\infty$

$x=\frac{1}{1-a}$

$a=\frac{x-1}{x}$ ......... (1)

Similarly,

$b=\frac{y-1}{y}$ ......... (2)

$c=\frac{z-1}{z}$ ......... (3)

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

Therefore,

$2b=a+c$

$2\times \frac{y-1}{y}=\frac{x-1}{x}+\frac{z-1}{z}$

$(2y-2)\left(xz\right)=xyz-zy+xyz-xy$

$2xyz-2xz=2xyz-xy-zy$

$2xz=y(x+z)$

$y=\frac{2xz}{x+z}$

Therefore, $x,y,z$ are in $H.P$.