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

If ${a_1}, {a_2}, {a_3}, \dots, {a_n}$ are in $H.P.$,then the expression ${a_1}{a_2} + {a_2}{a_3} + \dots + {a_{n - 1}}{a_n}$ is equal to:

If $\sin (\theta-\alpha), \sin \theta$ and $\sin (\theta+\alpha)$ are in $H.P.$,then the value of $\cos ^2 \theta$ is

Let $a_1, a_2, a_3, \ldots$ be in harmonic progression with $a_1 = 5$ and $a_{20} = 25$. The least positive integer $n$ for which $a_n < 0$ is

If for the harmonic progression,$t_{7} = \frac{1}{10}$ and $t_{12} = \frac{1}{25}$,then $t_{20} =$

An aeroplane flies around a square,the sides of which measure $100$ miles each. The aeroplane covers the first side at a speed of $100 \text{ mph}$,the second side at $200 \text{ mph}$,the third side at $300 \text{ mph}$,and the fourth side at $400 \text{ mph}$. The average speed of the aeroplane around the square is ....... $\text{mph}$.