Question

which explicit rule can be used to represent the sequence?
$a_n = 1 + (n - 1)4.5$
$a_n = 3 + (n - 1)1.5$
$a_n = 4.5 + (n - 1)1.5$
$a_n = 4.5 + (n + 1)1.5$

Explanation:

Step1: Identify the first term and common difference

The first term \(a_1\) from the point \((1, 4.5)\) is \(4.5\). To find the common difference \(d\), we can subtract consecutive terms. For example, \(a_2 - a_1 = 6 - 4.5 = 1.5\), \(a_3 - a_2 = 7.5 - 6 = 1.5\), so \(d = 1.5\).

Step2: Recall the explicit formula for an arithmetic sequence

The explicit formula for an arithmetic sequence is \(a_n = a_1 + (n - 1)d\).

Step3: Substitute \(a_1\) and \(d\) into the formula

Substituting \(a_1 = 4.5\) and \(d = 1.5\) into the formula, we get \(a_n = 4.5 + (n - 1)1.5\).

Answer:

\(a_n = 4.5 + (n - 1)1.5\) (the third option)