For all $n \in N$,which of the following is true: $\frac{3^n-1}{2} \geq$ ?

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

If three unequal numbers $p, q, r$ are in $H.P.$ and their squares are in $A.P.$,then the ratio $p:q:r$ is

If $A_1, A_2$ are two arithmetic means,$G_1, G_2$ are two geometric means,and $H_1, H_2$ are two harmonic means between two numbers,then $\frac{A_1 + A_2}{H_1 + H_2} \cdot \frac{H_1 H_2}{G_1 G_2} = \dots$

If three unequal non-zero real numbers $a, b, c$ are in $G.P.$ and $b - c, c - a, a - b$ are in $H.P.$,then the value of $a + b + c$ is independent of