If x=_n=0^a^n, y=_n=0^b^n, z=_n=0^c^n where a,b,c are in A.P.

Question

If $x=\displaystyle\sum_{n=0}^{\infty}a^n,\ y=\sum_{n=0}^{\infty}b^n,\ z=\sum_{n=0}^{\infty}c^n$ where $a,b,c$ are in A.P. and $|a|<1,|b|<1,|c|<1$, then $x,y,z$ are in:

Accepted Answer

$$x = \sum\limits_{n = 0}^\infty  {{a^n}}  = {1 \over {1 - a}}\,\,\,\,\,\,\,\,\,\,a = 1 - {1 \over x}$$

$$y = \sum\limits_{n = 0}^\infty  {{b^n}}  = {1 \over {1 - b}}\,\,\,\,\,\,\,\,\,\,b = 1 - {1 \over y}$$

$$z = \sum\limits_{n = 0}^\infty  {{c^n}}  = {1 \over {1 - c}}\,\,\,\,\,\,\,\,\,\,c = 1 - {1 \over z}$$

$a,b,c$ are in $A.P.$ OR $2b = a + c$

$$2\left( {1 - {1 \over y}} \right) = 1 - {1 \over x} + 1 - {1 \over y}$$

$${2 \over y} = {1 \over x} + {1 \over z} \Rightarrow x,y,z$$ are in $H.P.$

Solutions · 1