## 9. Summary

In summary, we have studied half-life to mathematically describe the decay of a substance. In particular, we considered a discrete-time model showing drug levels in the body. The discrete-time model was useful in that it allowed us to consider sequences as the basis for our model. Not only are sequences easy to work with mathematically, but they are also easy to graph and use for calculations. After considering our problem, we were able to formulate the iterative form of our sequence. Although the iterative form is useful, we then considered the general term of the sequence. The general term allowed us to quickly calculate a term of the sequence while avoiding many tedious calculations.

Using the simulator, we were able to visualize the discrete-time model we developed for numerous drugs. Our question of whether a drug is toxic or not is easily answered by using the simulator to compare the model with the toxic level. In turn, we looked at the long-term behaviour of our model, and noted that the drug levels in the body approach some steady state. By using simple algebra or limits from calculus, we found an expression for the steady state in terms of the dosage and the parameter .

Ultimately, we have found answers to our guiding questions, and we have developed a mathematical model describing drug levels in the body.

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