Write the first six terms of the following sequence:

$a_1=-1, a_n=\frac{a_{n-1}}{n}, \left(n\ge 2\right)$

Write the first six terms of the following sequence:

$a_1=4, a_{n+1}=2n{a}_n$

Write the first six terms of the following sequence:

$a_1=\frac{1}{2}, a_2=-1, a_{n+2}=a_n{a}_{n+1}$

Write the first six terms of the following sequences:

$a_1=a_2=2, a_n=a_{n-1}-1, \left(n>2\right)$

The sequence $a\left(n\right)$ is defined by: $a\left(n\right)=(n-1)(n-2)(n-3)$. Show that the first three terms of the sequence are zero, but the rest of the terms are positive.

The Fibonacci sequence is defined by:

$a_1=1=a_2, a_n=a_{n-1}+a_{n-2} \left(n>2\right)$

Find $\frac{a_{n+1}}{a_n}$, for $n=1, 2, 3, 4, 5$

Find the ${21}^{st}$ and ${42}^{nd}$ terms of the sequence defined by:

$t_n=\left\{\begin{array}{l}0, \mathrm{if} n \mathrm{is} \mathrm{odd}\\ 1, \mathrm{if} n \mathrm{is} \mathrm{even}\end{array}\right.$

Consider the sequence defined by: $t_n=a{n}^2+bn+c$

If $t_2=3, t_4=13$ and $t_7=113$, show that: $3t_n=17n^2-87n+115$

In the following situation, does the list of numbers involved make an arithmetic progression and why?

Number of students left in the school auditorium from the total strength of $800$ students when they leave the auditorium in batches of $20$.