If $a, b, c$ are in $A.P.$ and $|a|, |b|, |c| < 1$,and $x = 1 + a + a^2 + \dots \infty$,$y = 1 + b + b^2 + \dots \infty$,$z = 1 + c + c^2 + \dots \infty$,then $x, y, z$ shall be in:

If $a, b, c$ are in $A.P.$,$b, c, d$ are in $G.P.$,and $c, d, e$ are in $H.P.$,then $a, c, e$ are in

If the arithmetic mean of two numbers $a$ and $b$,where $a > b > 0$,is five times their geometric mean,then $\frac{a + b}{a - b}$ is equal to

Let $A_1, G_1, H_1$ be the arithmetic,geometric,and harmonic means of two distinct positive numbers. For $n \geq 2$,let $A_n, G_n, H_n$ be the arithmetic,geometric,and harmonic means of $A_{n-1}$ and $H_{n-1}$ respectively. Which of the following statements is true?

An $A.P.$,a $G.P.$,and a $H.P.$ have the same first and last terms and the same odd number of terms. The middle terms of the three series are in

The $8^{\text{th}}$ common term of the series $S_1 = 3 + 7 + 11 + 15 + 19 + \dots$ and $S_2 = 1 + 6 + 11 + 16 + 21 + \dots$ is $.......$.