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

There are two such pairs of non-zero real values of $a$ and $b$,i.e.,$(a_1, b_1)$ and $(a_2, b_2)$,for which $2a+b, a-b, a+3b$ are three consecutive terms of a $G.P.$. Then the value of $2(a_1b_2 + a_2b_1) + 9a_1a_2$ is-

If the $p^{th}$,$q^{th}$,and $r^{th}$ terms of an arithmetic sequence are $a$,$b$,and $c$ respectively,then the value of $[a(q - r) + b(r - p) + c(p - q)]$ is:

The mean of the series $a, a + nd, a + 2nd$ is

If $n$ arithmetic means $a_1, a_2, \dots, a_n$ are inserted between $50$ and $100$ and $n$ harmonic means $h_1, h_2, \dots, h_n$ are inserted between the same two numbers,then $a_2 h_{n-1}$ is equal to

If $\alpha, \beta$ are the roots of the equation $x^2 - 3x + a = 0$ and $\gamma, \delta$ are the roots of the equation $x^2 - 12x + b = 0$,and $\alpha, \beta, \gamma, \delta$ form an increasing $G.P.$,then $(a, b) = $