# Solving a difference equation

A certain smoker has a daily intake of 0.02 milligrams of nicotine. It is assumed that 1% of nicotine is disintegrated by the body per day.

Set up a difference equation for the amount of nicotine N_t after t days, starting with an initial level of N_0=0.

Derive a closed form of the solution for N_t.

If a concentration of 1mg of nicotine is considered harmful, when does the smoker reach this threshold? How much higher does the concentration rise?

#### Solution Preview

Let's consider the change in N_t during one day. We know that 0.02 milligrams is added each day, but also that 1% will vanish due to nicotine being broken down. We thus have:

N_{t+1} - N_{t} = 0.02 milligrams - 0.01 N_{t}

Of course, a small part fo the 0.02 mg also is disintegrated already during the day. If the 0.02 mg were added at the start f the day, we should replace it by 0.02*.99 mg, but it could be that the smoker smokes late in the evening. We will simply assume the net added amount of nicotine due to smoking after one day, talking into account that part of it gets destroyed, is 0.02 mg.

The above difference equation can be written as:

N_{t+1} - 0.99 N_{t} = 0.02 mg

The question is then how to solve such a difference equation. There are many different methods, let's use one of the most powerful methods: Generating functions. To avoid confusion with the variables, Let's rewrite the difference equation as:

a_{n+1} - s a_{n} = p (1)

a_{n} is now the amount of nicotine in the body after n days, s = 0.99 and p = 0.02 mg. We define the function f(x) as:

f(x) = Sum from n = 0 to infinity of a_{n} x^{n}

The variable x and the function f(x) don't have any meaning, they are purely formal mathematical objects. The idea is now to use the ...

#### Solution Summary

We explain how to set up a difference equation for the given problem and solve it using the method of generating functions. We explain this method in detail.