These are the first four terms of a sequence. 5/2 20/3 45/4 80/5

1. Recognise and extend the numerator and denominator patterns.

The terms are
$\frac{5}{2},\ \frac{20}{3},\ \frac{45}{4},\ \frac{80}{5}.$
The numerators are
$5,20,45,80=5(1^2),5(2^2),5(3^2),5(4^2).$
The denominators are
$2,3,4,5.$
So the next two terms are
$\frac{5(5^2)}{6}=\frac{125}{6},\qquad \frac{5(6^2)}{7}=\frac{180}{7}.$

1. Identify the numerator and denominator patterns and form the nth term.

The terms are
$\frac{5}{2},\ \frac{20}{3},\ \frac{45}{4},\ \frac{80}{5}.$
The numerators are
$5,\ 20,\ 45,\ 80=5(1^2),\ 5(2^2),\ 5(3^2),\ 5(4^2),$
so the numerator of the $n$th term is $5n^2$.

The denominators are
$2,\ 3,\ 4,\ 5,$
so the denominator of the $n$th term is $n+1$.

Therefore, the $n$th term is
$\frac{5n^2}{n+1}.$

1. Use the nth term to check whether the given value fits the sequence pattern.

From part (b), the $n$th term is
$\dfrac{5n^2}{n+1}$.

$71\dfrac{3}{16}=\dfrac{1139}{16}$.

If the denominator is $16$, then
$n+1=16 \Rightarrow n=15$.

For $n=15$,
$\dfrac{5n^2}{n+1}=\dfrac{5(15)^2}{16}=\dfrac{1125}{16}$.

Since $\dfrac{1125}{16}\ne\dfrac{1139}{16}$, $71\dfrac{3}{16}$ is not a term.