If $\frac{b + a}{b - a} = \frac{b + c}{b - c}$,then $a, b, c$ are in

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

For a positive integer $n$,let $a(n) = 1 + \frac{1}{2} + \frac{1}{3} + \frac{1}{4} + \dots + \frac{1}{2^n - 1}$. Then:

If $9, x, y, z, a$ are in $A.P.$ such that $x + y + z = 15$,and $9, x, y, z, a$ are in $H.P.$ such that $\frac{1}{x} + \frac{1}{y} + \frac{1}{z} = \frac{5}{3}$,then the value of $a$ is:

Let $A_1$ and $A_2$ be two arithmetic means and $G_1, G_2, G_3$ be three geometric means between two distinct positive numbers $a$ and $b$. Then $G_1^4 + G_2^4 + G_3^4 + G_1^2 G_3^2$ is equal to

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