6. Finding the General Term

The sequence, with

is the recursive definition of the sequence, meaning that each term in the sequence is calculated based on the previous term. Although iterative form is useful for answer some questions, it is still necessary to generalize the sequence such that its formulation does not depend on any previous terms. In this way, we can develop a formula that depends only on the characteristics of the drug: half-life, pill size, dosage, etc. The generalized sequence is useful in that we will use it to determine if a steady state exists and to simplify calculations.

Firstly, we know that each term is defined by the previous term. For instance, we have and .

Using this knowledge and starting with the recursive definition of our sequence, we have

In the first step, we replaced with , and simplified. If we continue this process, replacing and simplifying, we find a pattern:

 

A geometric series is a series with a common geometric ratio between terms. Starting with the first term , we have terms of a geometric series:

If , then the sum of terms of a geometric series is given by

 

For the underlined part of the last line above,

answer the following questions:

How many terms does the underlined series have?

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What is the common ratio?

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What is the first term?

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What is the sum of the underlined series?

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Now that we have simplified the underlined part, we can write our sequence in terms of , and as desired: