For every natural number $n$,$n(n + 1)$ is always

If the $(m + 1)^{th}$,$(n + 1)^{th}$ and $(r + 1)^{th}$ terms of an $A.P.$ are in $G.P.$ and $m, n, r$ are in $H.P.$,then the value of the ratio of the common difference to the first term of the $A.P.$ is

If $e^{(\cos^{2} x + \cos^{4} x + \cos^{6} x + \dots \infty) \log_{e} 2}$ satisfies the equation $t^{2} - 9t + 8 = 0$,then the value of $\frac{2 \sin x}{\sin x + \sqrt{3} \cos x}$ for $0 < x < \frac{\pi}{2}$ is

An arithmetic progression is written in the following way. The sum of all the terms of the $10^{\text{th}}$ row is..........

Let $a_1, a_2, a_3, \dots$ be an $A$.$P$. and $g_1, g_2, g_3, \dots$ be an increasing $G$.$P$. If $a_1 = g_1$ and $a_2 + g_2 = 1$ and $a_3 + g_3 = 4$, then $a_{10} + g_5$ is equal to:

If $a_1 = a_2 = 2$ and $a_n = a_{n-1} - 1$ for $n > 2$,then $a_5$ is