The sum of all $3$-digit numbers less than or equal to $500$,formed without using the digit $1$,which are also multiples of $11$,is ..... .

Let $n \geq 4$ be a positive integer and let $l_1, l_2, \ldots, l_n$ be the lengths of the sides of an arbitrary $n$-sided non-degenerate polygon $P$. Suppose $\frac{l_1}{l_2} + \frac{l_2}{l_3} + \ldots + \frac{l_{n-1}}{l_n} + \frac{l_n}{l_1} = n$. Consider the following statements:
$I$. The lengths of the sides of $P$ are equal.
$II$. The angles of $P$ are equal.
$III$. $P$ is a regular polygon if it is cyclic.

Let the sum of an infinite $G.P.$,whose first term is $a$ and the common ratio is $r$,be $5$. Let the sum of its first five terms be $\frac{98}{25}$. Then the sum of the first $21$ terms of an $A.P.$,whose first term is $10ar$,$n^{\text{th}}$ term is $a_n$ and the common difference is $10ar^2$,is equal to.

If $x=\sum_{n=0}^{\infty} \cos ^{2 n} \theta$,$y=\sum_{n=0}^{\infty} \sin ^{2 n} \theta$,$z=\sum_{n=0}^{\infty} \cos ^{2 n} \theta \sin ^{2 n} \theta$ and $0 < \theta < \frac{\pi}{2}$,then

If $a, b, c$ are in $G.P.$ and $\log a - \log 2b, \log 2b - \log 3c$ and $\log 3c - \log a$ are in $A.P.$,then $a, b, c$ are the lengths of the sides of a triangle which is

If $\log _e a, \log _e b, \log _e c$ are in an $A.P.$ and $\log _e a - \log _e 2b, \log _e 2b - \log _e 3c, \log _e 3c - \log _e a$ are also in an $A.P.$,then $a : b : c$ is equal to