Question

If $x=\displaystyle\sum_{n=0}^{\infty} a^{n}, y=\displaystyle\sum_{n=0}^{\infty} b^{n}, z=\displaystyle\sum_{n=0}^{\infty} c^{n}$ 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)

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