## 5.2 Formulating the Sequence: Part 2

In this section, we will devote our efforts to determining the mathematical relationship between sequence values. In other words, we will formulate our sequence mathematically!

Visually, we have points , , and below in Figure 12.

We wish to determine the relation between and .

How are and related?

What is the relation between and ?

Then the relationship between and is:

This relation was obtained by substituting the expression for into the expression for .

Now that we have the relation between and , we add points and to the diagram:

Again, we seek to find the relation between and . As before, and that and are related by decay.

We have the same as before:

When we continue this process, we develop our sequence. Each point is related to the previous in that you take the decay of the previous value over some interval of time and then add another dose.

In other words, given some initial dose, , we have

Does this formulation of the sequence make sense?

Yes! The sequence does make sense. We start with the amount . Now we ask: "what is ?" We relate to , in the same manner as we related to above.

Note that in our sequence, is a constant, as we are given and . For simplicity, we will call it , that is,

We can define our sequence formally: for some initial dose, , we have

Congratulations, we have formulated our sequence!

Note that which in this case is since we are assuming that there is no drug in the body before the initial dose, so the initial dose is just the size of the dose .

Consider a drug where hours, hours, and mg.

Find the next 3 terms of the sequence.

In the next section, we will look at an alternative form of the sequence—one that avoids tedious calculations like the those above.