已知一个数列 $x_1, x_2, x_3, x_4, \dots$ 的前四项如下。 $x_1 = 2(0) + 1 = 1$ $x_2 = 2(1) + 3 = 5$ $x_3 = 2(2) + 5 = 9$ $x_4 = 2(3) + 7 = 13$

1. Continue the given pattern to the fifth term

From the given terms,
$x_1=2(0)+1,\quad x_2=2(1)+3,\quad x_3=2(2)+5,\quad x_4=2(3)+7.$
So the next expression is
$x_5=2(4)+9.$

1. Express the pattern in terms of $n$

From the given terms,

$x_1=2(0)+1$, $x_2=2(1)+3$, $x_3=2(2)+5$, $x_4=2(3)+7$.

For the $n^{th}$ term, the number in brackets is $n-1$, and the odd number is $2n-1$.

So
$x_n=2(n-1)+(2n-1).$

2. Simplify the expression for the $n^{th}$ term

$x_n=2(n-1)+(2n-1)$
$=2n-2+2n-1$
$=4n-3.$

1. Substitute $n=20$ into the formula for the sequence and evaluate

Using $x_n=4n-1$,
$x_{20}=4(20)-1=80-1=79$

1. Test whether 203 can be written in the form of the nth term

From part (b),
$x_n=4n-1.$
If $203$ is a term, then
$4n-1=203.$

2. Solve for n and interpret whether it gives a valid term number

$4n=204$
$n=51$
Since $51$ is a positive integer, $x_{51}=203$.