Question

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

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

Answer 1

The correct option is C H.P.

$x,y,z$ are the sums of three different infinite G.P.'s

$x={\sum }_{n=0}^{\infty }{a}^n=\frac{1}{1-a}$, as $|a|<1$

$y={\sum }_{n=0}^{\infty }{b}^n=\frac{1}{1-b}$, as $|b|<1$

$z={\sum }_{n=0}^{\infty }{c}^n=\frac{1}{1-c}$, as $|c|<1$

If $a,b,c$ are in A.P., then $-a,-b,-c$ are also in A.P. and also $1-a,1-b,1-c$ are in A.P.

$\therefore \frac{1}{1-a},\frac{1}{1-b},\frac{1}{1-c}$ are in H.P.

So, $x,y,z$ are in H.P.

Hence, Option C is the correct answer.