What is the arithmetic mean of the series $a + (a + d) + (a + 2d) + \dots + (a + 2nd)$?

If $a, b, c, d, e, f$ are in an arithmetic progression,then $e - c = \dots$

If the sum of $n$ terms of an $A.P.$ is $S_n = nP + \frac{1}{2}n(n-1)Q$,where $P$ and $Q$ are constants,find the common difference.

The sides of a right-angled triangle are in arithmetic progression. If the triangle has an area of $24$,then what is the length of its smallest side?

$A$ software company sets up $m$ number of computer systems to finish an assignment in $17$ days. If $4$ computer systems crashed on the start of the second day,$4$ more computer systems crashed on the start of the third day and so on,then it took $8$ more days to finish the assignment. The value of $m$ is equal to :

If the sum of the first $n$ terms of an $A.P.$ is $cn(n - 1)$,where $c \neq 0$,then the sum of the squares of these terms is: