Question

after rogers surgery, his doctor administered pain medication in the following amounts in milligrams over four days.

day (n)	1	2	3	4

Explanation:

1. First, check if it is an arithmetic - sequence:

• The common difference \(d\) of an arithmetic sequence is \(d = a_{n + 1}-a_{n}\).
• \(d_1=1680 - 2000=-320\), \(d_2 = 1411.2-1680=-268.8\). Since \(d_1

eq d_2\), it is not an arithmetic sequence.

1. Then, check if it is a geometric sequence:

• The common ratio \(r\) of a geometric sequence is \(r=\frac{a_{n + 1}}{a_{n}}\).
• \(r_1=\frac{1680}{2000}=0.84\).
• \(r_2=\frac{1411.2}{1680}=0.84\).
• \(r_3=\frac{1185.4}{1411.2}=0.84\).
• Since the ratio between consecutive terms is constant (\(r = 0.84\)), this sequence can be modeled as a geometric sequence.
• The general formula for a geometric sequence is \(a_n=a_1r^{n - 1}\), where \(a_1 = 2000\) (the dosage on day 1) and \(r = 0.84\). So the model for the dosage \(m\) on day \(n\) is \(m = 2000\times(0.84)^{n - 1}\).

Step1: Check for arithmetic - sequence

Calculate differences between consecutive terms.
\(1680 - 2000=-320\), \(1411.2-1680=-268.8\)

Step2: Check for geometric - sequence

Calculate ratios between consecutive terms.
\(\frac{1680}{2000}=0.84\), \(\frac{1411.2}{1680}=0.84\), \(\frac{1185.4}{1411.2}=0.84\)

Step3: Find the geometric - sequence formula

Use \(a_n=a_1r^{n - 1}\) with \(a_1 = 2000\) and \(r = 0.84\).
\(m = 2000\times(0.84)^{n - 1}\)

Answer:

The sequence can be modeled as a geometric sequence with the formula \(m = 2000\times(0.84)^{n - 1}\)