If the sides of a right-angled triangle are in $A.P.$,then the sides are proportional to

The number of terms of the $A.P. 3, 7, 11, 15, ...$ to be taken so that the sum is $406$ is

The maximum sum of the series $20 + 19\frac{1}{3} + 18\frac{2}{3} + \dots$ is

If $n$ is a natural number,then $\left( \frac{n+1}{2} \right)^n \ge n!$ is true when

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:

The four arithmetic means between $3$ and $23$ are.....