Write the first three terms in each of the following sequences defined by the following: $a_{n} = \frac{n-3}{4}$

Let $S_{n} = \frac{1}{2} + \frac{1}{6} + \frac{1}{12} + \frac{1}{20} + \ldots$ up to $n$ terms. If the sum of the first six terms of an $A.P.$ with first term $-p$ and common difference $p$ is $\sqrt{2026 S_{2025}}$,then the absolute difference between the $20^{\text{th}}$ and $15^{\text{th}}$ terms of the $A.P.$ 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.

The sum of the first ten terms of an $A$.$P$. is $160$ and the sum of the first two terms of a $G$.$P$. is $8$. If the first term of the $A$.$P$. is equal to the common ratio of the $G$.$P$. and the first term of the $G$.$P$. is equal to the common difference of the $A$.$P$.,then the sum of all possible values of the first term of the $G$.$P$. is:

If $a_n = \sqrt{7+\sqrt{7+\sqrt{7+\ldots}}}$ ($n$ times),then which one of the following is true?

If the value of $\left(1+\frac{2}{3}+\frac{6}{3^{2}}+\frac{10}{3^{3}}+\ldots \text{ to } \infty\right)^{\log_{(0.25)}\left(\frac{1}{3}+\frac{1}{3^{2}}+\frac{1}{3^{3}}+\ldots \text{ to } \infty\right)}$ is $l$,then $l^{2}$ is equal to $......$