Determine $k$ so that $\frac{2}{3}, k$ and $\frac{5}{8} k$ are the three consecutive terms of an $A.P.$

The geometric series $a + ar + ar^2 + ar^3 + \dots \infty$ has a sum of $7$,and the sum of the terms involving odd powers of $r$ is $3$. Then,the value of $(a^2 - r^2)$ is:

If $b$ is the first term of an infinite $G.P.$ whose sum is $5$,then $b$ lies in the interval

The ratio of the sums of the first $n$ even numbers and $n$ odd numbers is

$A$ person is to count $4500$ currency notes. Let $a_n$ denote the number of notes he counts in the $n^{th}$ minute. If $a_1 = a_2 = \ldots = a_{10} = 150$ and $a_{10}, a_{11}, \ldots$ are in an $A.P.$ with common difference $-2$,then the time taken by him to count all notes is ............... minutes.

The $G.M.$ of the numbers $3, 3^2, 3^3, ..., 3^n$ is