$A$ 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?

Write the first three terms in each of the following sequences defined by the following:
$a_{n} = 2n + 5$

Three positive numbers $a, b,$ and $c$ are in an Arithmetic Progression $(AP)$ and $abc = 4$. The minimum possible value of $b$ is:

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

If the sum of $n$ terms of an $A.P.$ is $3n^{2} + 5n$ and its $m^{\text{th}}$ term is $164$,find the value of $m$.

Consider two sets $A$ and $B$,each containing three numbers in $A.P.$ Let the sum and the product of the elements of $A$ be $36$ and $p$ respectively,and the sum and the product of the elements of $B$ be $36$ and $q$ respectively. Let $d$ and $D$ be the common differences of the $A.P.s$ in $A$ and $B$ respectively such that $D = d + 3$ and $d > 0$. If $\frac{p + q}{p - q} = \frac{19}{5}$,then $p - q$ is equal to: