The value of the expression $3(1!) - 4(2!) + 5(3!) - 6(4!) + \dots - 2008(2006)! + (2007)!$ is:

When the $9^{th}$ term of an $A.P.$ is divided by its $2^{nd}$ term,the quotient is $5$. When the $13^{th}$ term is divided by the $6^{th}$ term,the quotient is $2$ and the remainder is $5$. Find the first term of the $A.P.$

The minimum value of ${\left( {\frac{3}{a} - 1} \right)^2} + {\left( {\frac{a}{b} - 1} \right)^2} + {\left( {\frac{b}{c} - 1} \right)^2} + {\left( {3c - 1} \right)^2}$ where $0 < a, b, c \leqslant 9$,is $p - q\sqrt{r}$; $p, q, r \in I$ and $q, r$ are coprime,then $(p + q + r)$ is equal to

The sum of three numbers in a $GP$ is $105$ and their product is $8000$. Find the numbers.

The common difference of the $A.P.$ $b_{1}, b_{2}, \ldots, b_{m}$ is $2$ more than the common difference of $A.P.$ $a_{1}, a_{2}, \ldots, a_{n}.$ If $a_{40} = -159,$ $a_{100} = -399$ and $b_{100} = a_{70},$ then $b_{1}$ is equal to

$a, g, h$ are the arithmetic mean,geometric mean,and harmonic mean between two positive numbers $x$ and $y$ respectively. Identify the correct statement among the following: