If $b + c, c + a, a + b$ are in $H.P.$,then $\frac{a}{b + c}, \frac{b}{c + a}, \frac{c}{a + b}$ are in

If the harmonic mean between $a$ and $b$ is $H$,then $\frac{H + a}{H - a} + \frac{H + b}{H - b} = $

Let $a_1, a_2, a_3, \dots$ be a harmonic progression where $a_1 = 5$ and $a_{20} = 25$. What is the smallest positive integer $n$ such that $a_n < 0$?

If $\cos (\theta-\alpha), \cos \theta$ and $\cos (\theta+\alpha)$ are in harmonic progression,then $2 \tan ^2 \theta=$

The harmonic mean of $\frac{a}{1 - ab}$ and $\frac{a}{1 + ab}$ is $......$

If there are $n$ harmonic means between $1$ and $\frac{1}{31}$ and the ratio of the $7^{th}$ and $(n - 1)^{th}$ harmonic means is $9:5$,then the value of $n$ is: