Question

Let $a,b,c$ be in $AP$. If $0<a,b,c<1,x=\sum _{n=0}^{\infty }{a}^n,$ $y=\sum _{n=0}^{\infty }{b}^n$ and $z=\sum _{n=0}^{\infty }{c}^n,$ then

• A. $2y=x+z$
• B. $2x=y+z$
• C. $2z=x+y$
• D. $2xz=xy+yz$
• E. $z=\frac{2xy}{x+y}$

Answer 1

Answer: (D)

Since, $x=\sum _{n=0}^{\infty }{a}^n$
$\therefore$ $x=1+a+a^2+....\infty$
$\Rightarrow$ $x=\frac{1}{1-a}$
$\Rightarrow$ $(1-a)x=1$
$\Rightarrow$ $a=\frac{x-1}{x}$ Similarly, $b=\frac{y-1}{y}$ and $c=\frac{z-1}{z}$
Since, a, b and c are in AP.
$\therefore$ $b=\frac{a+c}{2}$
$\Rightarrow$ $\frac{y-1}{y}=\frac{\frac{x-1}{x}+\frac{z-1}{z}}{2}$
$\Rightarrow$ $2xz(y-1)=y[z(x-1)+x(z-1)]$
$\Rightarrow$ $2xyz-2xz=xyz-yz+xyz-xy$
$\Rightarrow$ $-2xz=-yz-xy$
$\Rightarrow$ $2xz=xy+yz$