Five numbers are in an $AP$ with a common difference $d \neq 0$. If the $1^{st}$, $3^{rd}$, and $4^{th}$ terms are in a $GP$, then:

Define a sequence $\langle a_n \rangle$ by $a_1 = 5, a_n = a_1 a_2 \dots a_{n-1} + 4$ for $n > 1$. Then,$\lim_{n \to \infty} \frac{\sqrt{a_n}}{a_{n-1}}$

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.

If three unequal non-zero real numbers $a, b, c$ are in $G.P.$ and $b - c, c - a, a - b$ are in $H.P.$,then the value of $a + b + c$ is independent of

If $\log_x y, \log_z x, \log_y z$ are in $G.P.$,$xyz = 64$,and $x^3, y^3, z^3$ are in $A.P.$,then

Let the sum of the first $n$ terms of an $A$.$P$. be $3n^2 + 5n$. Then the sum of squares of the first $10$ terms of the $A$.$P$. is: