Question

write a rule for the n th term of the sequence.
(5, 1.25)
(3, 0.75)
(4, 1)
(2, 0.5)
(1, 0.25)

$a_n = \square$

Explanation:

Step1: Identify sequence type

Check ratio between terms:
$\frac{0.5}{0.25}=2$, $\frac{0.75}{0.5}=1.5$, $\frac{1}{0.75}=\frac{4}{3}$, $\frac{1.25}{1}=1.25$. Ratios are not constant, so check linear relationship (difference between terms):
$0.5-0.25=0.25$, $0.75-0.5=0.25$, $1-0.75=0.25$, $1.25-1=0.25$. Common difference $d=0.25$, so it is an arithmetic sequence.

Step2: Recall arithmetic sequence formula

The formula for the nth term of an arithmetic sequence is $a_n = a_1 + (n-1)d$, where $a_1$ is the first term, $d$ is the common difference.

Step3: Substitute known values

$a_1=0.25$, $d=0.25$. Substitute into formula:
$a_n = 0.25 + (n-1)(0.25)$

Step4: Simplify the expression

$a_n = 0.25 + 0.25n - 0.25 = 0.25n$
Or in fraction form: $a_n = \frac{1}{4}n$

Answer:

$a_n = \frac{1}{4}n$ (or $a_n = 0.25n$)