If for the series $a_1, a_2, a_3, \ldots$, the expression $a_r - a_{r+1}$ bears a constant ratio with $a_r a_{r+1}$, then $a_1, a_2, a_3, \ldots$ are in:

If $a, b, c$ are in a geometric progression and $a^x = b^y = c^z$,then in which progression are $x, y, z$?

If $a, b, c$ are in $G$.$P$. $(a > 1, b > 1, c > 1)$, then for any real number $x$ $(x > 0, x \neq 1)$, the terms $\log_{a} x, \log_{b} x, \log_{c} x$ are in:

If the $p^{th}$ term of a harmonic progression is $qr$ and the $q^{th}$ term is $pr$,then what is its $r^{th}$ term?

Let the positive numbers $a, b, c, d$ be in $A.P.$,then $abc, abd, acd, bcd$ are

If $x = \sum_{n=0}^{\infty} a^n$,$y = \sum_{n=0}^{\infty} b^n$,and $z = \sum_{n=0}^{\infty} c^n$,where $a, b, c$ are in arithmetic progression and $|a| < 1, |b| < 1, |c| < 1$,then $x, y, z$ are in which progression?