Question

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

• A. $2y=x+z$
• B. $2x=y+z$
• C. $2z=x+y$
• D. $2xz=xy+yz$

Answer 1

Answer: (D)

$2xz=xy+yz$

Answer 2

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