Question

If $a,b,c$ are in $AP$ and $x,y,z$ are finite numbers such that $x=\sum _{n=0}^{\infty }{a}^n,y=\sum _{n=0}^{\infty }{b}^{n}andz=\sum _{n=0}^{\infty }{c}^n$ then $x,y,z$ are in

• A. $AP$
• B. $GP$
• C. $HP$
• D. $Arithmetico-geometricprogression$

Answer 1

The correct option is C $HP$

Given that, $x={\sum }_{n=0}^{\infty }{a}^n=\frac{1}{1-a}$
and $y={\sum }_{n=0}^{\infty }{b}^n=\frac{1}{1-b}$
and $z={\sum }_{n=0}^{\infty }{c}^n=\frac{1}{1-c}$
Since, $a$, $b$, $c$ are in A.P
$\therefore 1-a$, $1-b$, $1-c$ are in A.P
$\therefore \frac{1}{1-a}$, $\frac{1}{1-b}$, $\frac{1}{1-c}$ are in H.P
$\therefore x$, $y$, $z$ are in H.P
Ans: C