If the sum of the first four terms of an $A.P.$ is $6$ and the sum of its first six terms is $4$, then the sum of its first twelve terms is

If the $m^{th}$ term of an arithmetic progression is $1/n$ and the $n^{th}$ term is $1/m$,then the sum of the first $mn$ terms is:

$A$ man counts $4500$ currency notes. Let $a_n$ denote the number of notes he counts in the $n^{th}$ minute. If $a_1 = a_2 = \dots = a_{10} = 150$ and $a_{10}, a_{11}, \dots$ form an arithmetic progression with a common difference of $-2$,then how many minutes will it take for him to count all the notes?

The sum of numbers from $250$ to $1000$ which are divisible by $3$ is

The series of positive multiples of $3$ is divided into sets: $\{3\}, \{6, 9, 12\}, \{15, 18, 21, 24, 27\}, \ldots$. Then the sum of the elements in the $11^{\text{th}}$ set is equal to $................$

The $p^{th}$ term of the series $\left( 3 - \frac{1}{n} \right) + \left( 3 - \frac{2}{n} \right) + \left( 3 - \frac{3}{n} \right) + \dots$ is