Difference equations and recurrences describe how a quantity changes step by step instead of continuously. They are used in finance, population models, computer science, and many applied math problems. This cheat sheet helps students recognize common recurrence types, move between recursive and explicit forms, and interpret long-term behavior.
It is useful when a problem gives a starting value and a rule for finding the next value.
The most important ideas are first-order recurrences, arithmetic growth, geometric growth, and linear recurrences with a constant term. Students should know how to compute terms, write an explicit formula, and find an equilibrium when possible. Stability explains whether repeated steps move toward or away from an equilibrium.
Careful indexing matters because a small change in n can shift every term.
Key Facts
- A recurrence defines a sequence using earlier terms, such as a_n = a_(n-1) + 5 with an initial value a_0.
- An arithmetic recurrence has the form a_n = a_(n-1) + d, and its explicit form is a_n = a_0 + nd.
- A geometric recurrence has the form a_n = r a_(n-1), and its explicit form is a_n = a_0 r^n.
- A first-order linear recurrence with constant term has the form x_(n+1) = a x_n + b.
- For x_(n+1) = a x_n + b, the equilibrium value is x* = b/(1 - a), as long as a is not equal to 1.
- For x_(n+1) = a x_n + b, the explicit solution is x_n = x* + a^n(x_0 - x*) when a is not equal to 1.
- An equilibrium is stable when |a| < 1 because a^n approaches 0 as n increases.
- If |a| > 1, the recurrence x_(n+1) = a x_n + b moves away from equilibrium unless x_0 is exactly the equilibrium.
Vocabulary
- Recurrence relation
- A rule that defines each term of a sequence using one or more previous terms.
- Initial value
- The starting term needed to generate the rest of a sequence from a recurrence.
- Explicit formula
- A formula that gives a_n directly in terms of n without needing to compute earlier terms.
- Equilibrium
- A value that stays unchanged when the recurrence rule is applied.
- Stability
- The behavior of a recurrence near an equilibrium, especially whether terms move toward it or away from it.
- First-order difference equation
- A recurrence where the next value depends only on the current value, often written x_(n+1) = f(x_n).
Common Mistakes to Avoid
- Forgetting the initial value, which is wrong because a recurrence cannot generate a unique sequence without a starting term.
- Mixing up a_n = a_0 + nd and a_n = a_1 + nd, which is wrong because the index of the first term changes the count of steps.
- Using the geometric formula for an arithmetic recurrence, which is wrong because adding a fixed amount and multiplying by a fixed ratio create different growth patterns.
- Finding equilibrium by setting x_(n+1) equal to 0, which is wrong because equilibrium means x_(n+1) = x_n, not that the next value is zero.
- Calling every increasing recurrence stable, which is wrong because stability depends on whether terms move toward an equilibrium, not only whether they increase.
Practice Questions
- 1 Given a_0 = 7 and a_n = a_(n-1) + 4, find a_5 and write the explicit formula for a_n.
- 2 A savings account follows B_(n+1) = 1.03B_n + 200 with B_0 = 1000. Find B_2.
- 3 For x_(n+1) = 0.6x_n + 8, find the equilibrium value and state whether it is stable.
- 4 A population model has the rule P_(n+1) = 1.2P_n - 50. Explain what the multiplier 1.2 and the term -50 mean in context, and predict whether the model can show long-term growth.