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

Question

$x=\displaystyle\sum_{n=0}^{\infty}a^n$, $y=\displaystyle\sum_{n=0}^{\infty}b^n$, $z=\displaystyle\sum_{n=0}^{\infty}c^n$, where $a,b,c$ are in A.P. and $|a|<1,|b|<1,|c|<1$, $abc
e0$. Then:

Accepted Answer

$x=\dfrac{1}{1-a}$, $y=\dfrac{1}{1-b}$, $z=\dfrac{1}{1-c}$. Since $a,b,c$ are in A.P., so are $1-a,1-b,1-c$, meaning $x,y,z$ are in H.P., i.e. $\dfrac1x,\dfrac1y,\dfrac1z$ are in A.P.

Solutions · 1