Discrete Models
Core concepts
Discrete models involve recurrence relations, sequences, difference equations, first-order linear recurrences (t_n = a,t_{n-1}+b), and logistic growth with a carrying capacity. You must understand steady state / equilibrium solutions, iteration, graphing a recurrence, and modelling populations, loans and savings as recurrences. Also know arithmetic and geometric sequences as special cases.
Traps to watch
- Indexing off-by-one: (t_1) is the first term after the initial value (t_0); check which index the question uses.
- Steady state: solve (S = aS+b) to get (S=b/(1-a)) - this only exists when (|a|<1), otherwise the sequence diverges.
- Logistic recurrence: growth slows near the carrying capacity; do not assume exponential growth continues.
- Arithmetic vs geometric: arithmetic adds a constant; geometric multiplies by a constant.
- Using the wrong base in a savings recurrence - the interest multiplies the balance, then the deposit is added.
- Rounding iteratively - keep full precision and round only the final answer.
Worked rule of thumb
A recurrence (t_n = a,t_{n-1}+b) has steady state (S=\frac{b}{1-a}) when (|a|<1). Example: a savings account starts at $1000, earns 5% interest, and $200 is deposited each year. Then (t_n=1.05,t_{n-1}+200). Steady state: (S=200/(1-1.05)=-4000), which is negative and meaningless - the sequence diverges because (a>1), so the balance grows without bound. This is a useful sanity check: only apply the steady-state formula when (|a|<1).