Production equation
Each week a company produces X units and sells p per cent of its
stock. How should the company plan its warehouse space?
Problem
Each week a company produces $X$ units and sells $p$ per cent of its stock. How should the company plan its warehouse space? Will the stock fluctuate, or increase or decrease over time, or tend to a limit?
Initially the company has no stock. Show that, over a long period of time, the amount of stock tends to the limit ${100X\over p}$.
Getting Started
Taking $X_n$ as the amount of stock at the end of week $n$, you
need to solve the difference equation (recurrence relation)
Put $X_n = Y_n - C$ then
choose $C$ such that and
consider the values of this expression for $Y_n, Y_{n-1}, Y_{n-2},
... Y_1, Y_0$.
Student Solutions
Simon from Elizabeth College, Guensey and Andrei from Tudor Vianu National College, Romania both solved this problem using geometric series. Here is Simon's solution:
The equation to show stock levels in week $n$ is:
Writing $U_n$ in terms of $U_0$:
Teachers' Resources
The general method of solving problems where there is a formula for
the transition from one phase to the next is to solve either a
difference equation (such as this), or a differential equation,
according to whether the model is discrete or continuous.