The first four terms of a sequence are 27, 24, 21 and 18.

1. Find the common difference and continue the sequence to the 8th term.

The sequence decreases by $3$ each time:

$27,\ 24,\ 21,\ 18,\ 15,\ 12,\ 9,\ 6$

So the 8th term is $6$.

1. Find the common difference and form the nth term.

The sequence is $27, 24, 21, 18, \ldots$

Common difference:
$24-27=-3$

Using $T_n=a+(n-1)d$,
$T_n=27+(n-1)(-3)$
$T_n=27-3n+3=30-3n$

1. Check whether $-31$ can be produced by the nth term formula.

From part (b), the nth term is $30-3n$.

Set it equal to $-31$:
$30-3n=-31$
$-3n=-61$
$n=\frac{61}{3}$

Since $\frac{61}{3}$ is not a positive integer, $-31$ is not a term in the sequence.