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 $|a|<1,|b|<1,|c|<1$ then $x,y,z$ are in

• A. G. P.
• B. A.P.
• C. H.P.
• D. None of these

Answer 1

Answer: (C)

$x=\sum _{n=0}^{\infty }{a}^n=\frac{1}{1-a}$
$a=1-\frac{1}{x}$
$y=\sum _{n=0}^{\infty }{b}^n=\frac{1}{1-b}b=1-\frac{1}{y}$
$z=\sum _{n=0}^{\infty }{c}^n=\frac{1}{1-c}c=1-\frac{1}{z}$
$a,b,c$ are in A.P.
OR $2b=a+c$
$2\left(1-\frac{1}{y}\right)=1-\frac{1}{x}+1-\frac{1}{y}\frac{2}{y}=\frac{1}{x}+\frac{1}{z}$
$\Rightarrow x,y,z$ are in H.P