Let $I(n) = n^n$ and $J(n) = 1 \times 3 \times 5 \times \ldots \times (2n - 1)$ for all $n > 1, n \in N$. Then:

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

The value of $\left[\frac{2^{2020}+1}{2^{2018}+1}\right]+\left[\frac{3^{2020}+1}{3^{2018}+1}\right]+\left[\frac{4^{2020}+1}{4^{2018}+1}\right] +\left[\frac{5^{2020}+1}{5^{2018}+1}\right] + \left[\frac{6^{2020}+1}{6^{2018}+1}\right]$ is (where $[\cdot]$ denotes the greatest integer function):

Write the first five terms of the sequence defined by $a_{1} = -1$ and $a_{n} = \frac{a_{n-1}}{n}$ for $n \geq 2$,and obtain the corresponding series.

Let $\{a_{n}\}_{n=0}^{\infty}$ be a sequence such that $a_{0}=0, a_{1}=0$ and $a_{n+2}=3a_{n+1}-2a_{n}+1$ for all $n \geq 0$. Then the value of $a_{25}a_{23}-2a_{25}a_{22}-2a_{23}a_{24}+4a_{22}a_{24}$ is equal to:

For a natural number $n$,let $a_{n} = 19^{n} - 12^{n}$. Then,the value of $\frac{31 a_{9} - a_{10}}{57 a_{8}}$ is