某数列的前四项为 5, 9, 13 和 17。

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

The sequence is $5, 9, 13, 17, \ldots$

Each term increases by $4$.

Continuing:
$21,\ 25,\ 29,\ 33$

So the 8th term is $33$.

1. Identify the common difference of the sequence.

The sequence is $5, 9, 13, 17, \ldots$

Common difference:
$9-5=4$
So the terms increase by $4$ each time.

2. Form the expression for the nth term.

For an arithmetic sequence,
$T_n=\text{first term}+(n-1)(\text{common difference})$
$T_n=5+(n-1)(4)$
$T_n=5+4n-4=4n+1$

1. Substitute the given term value into the nth-term expression and solve for n.

From part (b), the nth term is $4n+1$.

Since the term is $205$,
$4n+1=205$
$4n=204$
$n=51$

1. Use the nth-term expression and equate it to 50.

From part (b), the nth term is $4n+1$.

If $50$ is a term of the sequence,
$4n+1=50.$

2. Solve for $n$ and justify why this means 50 is not in the sequence.

$4n=49$
$n=\frac{49}{4}=12.25$

Since $n$ must be a positive integer term number, $50$ is not part of the sequence.