Find the sum of all natural numbers lying between $100$ and $1000$ which are multiples of $5.$

$150$ workers were engaged to finish a piece of work in a certain number of days. $4$ workers dropped the second day,$4$ more workers dropped the third day and so on. It takes eight more days to finish the work now. The number of days in which the work was completed is

$a_1, a_2, a_3, \dots, a_{100}$ are in an arithmetic progression,where $a_1 = 3$ and $S_p = \sum_{i=1}^p a_i, 1 \le p \le 100$. For any integer $n$,let $m = 5n$. If $S_m/S_n$ is independent of $n$,then $a_2 = \dots$

If $a_1, a_2, a_3, \dots, a_n$ are in an arithmetic progression with common difference $d$,then find the value of $\sin d [\csc a_1 \csc a_2 + \csc a_2 \csc a_3 + \dots + \csc a_{n-1} \csc a_n]$.

Let the sum of the first three terms of an $A.P.$ be $39$ and the sum of its last four terms be $178.$ If the first term of this $A.P.$ is $10,$ then the median of the $A.P.$ is

If the sum of first $n$ terms of an $A.P.$ is equal to the sum of its first $m$ terms,$(m \ne n)$,then the sum of its first $(m + n)$ terms will be