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 . such that |a |<1, |b| > <1 and |c| < 1 then show that x , y , z are in H . P

Answer 1

$x=\frac{1}{1-a}\Rightarrow a=1-\frac{1}{x}by$Sum of infinity terms for G . P.

Similarly b,c are b=$1-\frac{1}{y}$,c=$1-\frac{1}{z}$
$\because$ a , b , c are in A . P . $\Rightarrow$ 2b = a + c
$\therefore 2(1-\frac{1}{y})=a-\frac{1}{x}+a-\frac{1}{z}\Rightarrow \frac{2}{y}=\frac{1}{x}+\frac{1}{z}\Rightarrow \frac{1}{x},\frac{1}{y},\frac{1}{z}$ are in the A . P . or x , y , z , are in H . P