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 AP and $\left|a\right|<1,\left|b\right|<1,\left|c\right|<1$ then x, y, z are in:

Answer 1

All three are sum of infinite G.P. with

$r$ ( multiply ratio) $<1\{|a|,|b|,|c|<1\}$

$\Rightarrow$ Sum of infinite G.P. $\sum _{n=0}^{\infty }{a}^n=\frac{1}{1-a}$

Sum of infinite G.P. $\sum b^n=\frac{1}{1-b},\sum c^n=\frac{1}{1-c}$

$\Rightarrow \frac{1}{x}=1-a,\frac{1}{y}=1-b,\frac{1}{z}=1-c$

$\Rightarrow \frac{1}{y}-\frac{1}{x}=a-b,\frac{1}{z}-\frac{1}{y}=b-c$

As $a,b,c$ are in $A.P,\Rightarrow a-b$ and $b-c$ are equal

$\Rightarrow \frac{1}{x},\frac{1}{y},\frac{1}{z}$ are in $A.P.$

$\Rightarrow x,y,z$ are in $H.P.$