The value of $x$ satisfying ${\log _a}x + {\log _{\sqrt a }}x + {\log _{3\sqrt a }}x + .........{\log _{a\sqrt a }}x = \frac{{a + 1}}{2}$ will be

The sum of $24$ terms of the following series $\sqrt 2 + \sqrt 8 + \sqrt {18} + \sqrt {32} + .........$ is

If $p,\;q,\;r$ are in $A.P.$ and are positive, the roots of the quadratic equation $p{x^2} + qx + r = 0$ are all real for

For a series $S = 1 -2 + 3\, -\, 4 … n$ terms,

Statement $-1$ : Sum of series always dependent on the value of $n$ , i.e. whether it is even or odd. 

Statement $-2$ : Sum of series is $-\frac {n}{2}$ when value of $n$ is any even integer

If $\log _e \mathrm{a}, \log _e \mathrm{~b}, \log _e \mathrm{c}$ are in an $A.P.$ and $\log _e \mathrm{a}-$ $\log _e 2 b, \log _e 2 b-\log _e 3 c, \log _e 3 c-\log _e a$ are also in an $A.P,$ then $a: b: c$ is equal to

The sums of $n$ terms of three $A.P.'s$ whose first term is $1$ and common differences are $1, 2, 3$ are ${S_1},\;{S_2},\;{S_3}$ respectively. The true relation is