Let $a_1, a_2, a_3, \ldots$ be an $A$.$P$. If $a_7 = 3$,the product $a_1 a_4$ is minimum and the sum of its first $n$ terms is zero,then $n! - 4 a_{n(n+2)}$ is equal to :

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 $......$

Let $a_n = \frac{10^n}{n!}$ for $n = 1, 2, 3, \ldots$. Then the greatest value of $n$ for which $a_n$ is the greatest is

The sum $1 \cdot 1^2 - 2 \cdot 3^2 + 3 \cdot 5^2 - 4 \cdot 7^2 + 5 \cdot 9^2 - \ldots + 15 \cdot 29^2$ is $.......$.

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

If the integers from $1$ to $2021$ are written as a single integer like $123 \dots 91011 \dots 20202021$,then the $2021^{st}$ digit (counted from the left) in the resulting number is