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):

If $x$ is the arithmetic mean and $y, z$ are the two geometric means between two positive numbers,then $\frac{y^3 + z^3}{xyz} = \dots..$

If $m$ arithmetic means $(A.Ms)$ and three geometric means $(G.Ms)$ are inserted between $3$ and $243$ such that the $4^{\text{th}}$ $A.M.$ is equal to the $2^{\text{nd}}$ $G.M.$,then $m$ is equal to:

Let $a_1, a_2, a_3, \ldots$ be a sequence of positive integers in arithmetic progression with common difference $2$. Also,let $b_1, b_2, b_3, \ldots$ be a sequence of positive integers in geometric progression with common ratio $2$. If $a_1 = b_1 = c$,then the number of all possible values of $c$,for which the equality $2(a_1 + a_2 + \ldots + a_n) = b_1 + b_2 + \ldots + b_n$ holds for some positive integer $n$,is:

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,$\lim_{n \to \infty} \frac{\sqrt{a_n}}{a_{n-1}}$

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.