Which number should be added to the numbers $13, 15, 19$ so that the resulting numbers are the consecutive terms of a $H.P.$?

Let $a, b, c, p, q$ and $r$ be positive real numbers such that $a, b$ and $c$ are in $GP$ and $a^{p} = b^{q} = c^{r}$. Then,

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 the roots of the equation $k x^3 - 18 x^2 - 36 x + 8 = 0$ are in harmonic progression,then $k =$

If $a, b, c$ are three distinct positive real numbers which are in $H.P.$,then $\frac{3a + 2b}{2a - b} + \frac{3c + 2b}{2c - b}$ is

If $H$ is the harmonic mean between $a$ and $b$,then what is the value of $\frac{1}{H - a} + \frac{1}{H - b}$?