If $(b+c), (c+a), (a+b)$ are in $H.P.$,then $a^2, b^2, c^2$ are in.......

The value of $x$ satisfying $\log_a x + \log_{\sqrt{a}} x + \log_{\sqrt[3]{a}} x + \dots + \log_{\sqrt[a]{a}} x = \frac{a(a+1)}{2}$ is:

If the geometric mean between $a$ and $b$ is $\frac{a^{n + 1} + b^{n + 1}}{a^n + b^n}$,then the value of $n$ is

Let $a_{1}, a_{2}, a_{3}, \ldots$ be a $G$.$P$. such that $a_{1} < 0$; $a_{1} + a_{2} = 4$ and $a_{3} + a_{4} = 16$. If $\sum_{i=1}^{9} a_{i} = 4 \lambda$,then $\lambda$ is equal to:

If $m$ is a root of the given equation $(1 - ab)x^2 - (a^2 + b^2)x - (1 + ab) = 0$ and $m$ harmonic means are inserted between $a$ and $b$,then the difference between the last and the first of the means equals

Let $a_1, a_2, a_3, ..., a_n$ be in an $A.P.$ If $a_3 + a_7 + a_{11} + a_{15} = 72$,then the sum of its first $17$ terms is equal to: