The sum of $24$ terms of the following series $\sqrt{2} + \sqrt{8} + \sqrt{18} + \sqrt{32} + \dots$ is

The number of $5$-tuples $(a, b, c, d, e)$ of positive integers such that:
$I.$ $a, b, c, d, e$ are the measures of angles of a convex pentagon in degrees.
$II.$ $a \leq b \leq c \leq d \leq e$.
$III.$ $a, b, c, d, e$ are in an arithmetic progression.

Let $S_n$ be the sum of the first $n$ terms of an arithmetic progression $3, 7, 11, \ldots$. If $40 < \left(\frac{6}{n(n+1)} \sum_{k=1}^{n} S_{k}\right) < 42$,then $n$ equals

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 first,second and last terms of an $A.P.$ are $a, b$ and $2a$ respectively,then its sum is:

Let $l_1, l_2, \ldots, l_{100}$ be consecutive terms of an arithmetic progression with common difference $d_1$,and let $w_1, w_2, \ldots, w_{100}$ be consecutive terms of another arithmetic progression with common difference $d_2$,where $d_1 d_2 = 10$. For each $i = 1, 2, \ldots, 100$,let $R_i$ be a rectangle with length $l_i$,width $w_i$,and area $A_i$. If $A_{51} - A_{50} = 1000$,then the value of $A_{100} - A_{90}$ is: