The least positive integer $n$ for which $\sqrt[3]{n+1}-\sqrt[3]{n} < \frac{1}{12}$ is

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

Let $x_1, x_2, \ldots, x_{100}$ be in an arithmetic progression,with $x_1 = 2$ and their mean equal to $200$. If $y_i = i(x_i - i)$ for $1 \leq i \leq 100$,then the mean of $y_1, y_2, \ldots, y_{100}$ is

Let $x_{1}, x_{2}, x_{3}, \ldots, x_{20}$ be in a geometric progression with $x_{1} = 3$ and common ratio $r = \frac{1}{2}$. $A$ new data set is constructed by replacing each $x_{i}$ with $(x_{i} - i)^{2}$. If $\bar{x}$ is the mean of the new data,then the greatest integer less than or equal to $\bar{x}$ is $.....$

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

For every natural number $n$,$n(n + 1)$ is always