--- title: "Recurrence Relations and Characteristic Roots" description: "Turn a rule that defines each term from earlier ones into a closed formula by solving the characteristic equation, including the repeated-root case." canonical: https://lightmysky.com/learn/mathematics/recurrence-relations-and-characteristic-roots-mt_q0b2bi0iAB source: https://lightmysky.com/learn/mathematics/recurrence-relations-and-characteristic-roots-mt_q0b2bi0iAB.md retrieved: 2026-09-12 --- > **Agent view.** This is the Markdown twin of the page, for tools and assistants. > When to use this site, and the call that answers each job: https://lightmysky.com/agent-instructions.md > API description (OpenAPI 3.1): https://lightmysky.com/openapi.json · Authentication: https://lightmysky.com/auth.md > Pricing: https://lightmysky.com/pricing.md · Catalog: https://lightmysky.com/llms.txt · Full catalog: https://lightmysky.com/llms-full.txt > Every machine-readable file on this domain: https://lightmysky.com/.well-known/ai-catalog.json > Ask for Markdown with `Accept: text/markdown`, a `.md` address, or `?mode=agent`. # Recurrence Relations and Characteristic Roots Turn a rule that defines each term from earlier ones into a closed formula by solving the characteristic equation, including the repeated-root case. Subject: Mathematics · Area: Discrete Mathematics · Ages 19 to 20 Page: https://lightmysky.com/learn/mathematics/recurrence-relations-and-characteristic-roots-mt_q0b2bi0iAB ## Ready when they can - Set up a linear recurrence from a described counting process - Solve a second-order homogeneous recurrence and fit the constants to the initial values - Handle a repeated characteristic root and explain why an extra factor of n appears ## Lesson: From a recurrence to a closed formula A recurrence defines each term from earlier ones, and two starting values get the whole chain going. The Fibonacci rule says each term is the sum of the two before it. A rabbit colony that doubles each month follows p(n) equals 2 times p(n minus 1), so 3 pairs grow as 3, 6, 12, 24, 48. Every recurrence needs both the rule and its seeds. Guessing r to the power n turns the rule into algebra for r. For a(n) equals 5 times a(n minus 1) minus 6 times a(n minus 2), dividing through gives r squared minus 5 r plus 6 equals 0. That factors as (r minus 2)(r minus 3), so the roots are 2 and 3. A geometric sequence works the same way: its ratio raised to n jumps straight to any term. **Example.** With distinct roots, blend the root powers and fit the blend to the start. The solution looks like A times 2 to the power n plus B times 3 to the power n. Each starting value gives one equation, so a0 and a1 pin down A and B together. Always check the formula against the first few terms: a mismatch at n = 2 rejects it. **Tip.** A repeated root supplies only one solution, so multiply by n for the second one. For (r minus 2) squared, the general form is A times 2 to the power n plus B times n times 2 to the power n. Two plain copies would collapse into one constant, leaving one degree of freedom short. The n factor is what makes the second solution independent. **Recap.** Write the rule, solve the characteristic equation, blend the root powers, and fit the blend to the start. ## Practice 14 questions on this page, each with its working shown. ## Needs first - [Proof by Induction and Strong Induction](https://lightmysky.com/learn/mathematics/proof-by-induction-and-strong-induction-mt_7lvC02JBOC) - [Second-Order Linear Equations with Constant Coefficients](https://lightmysky.com/learn/mathematics/second-order-linear-equations-with-constant-coefficients-mt_G7sMq1EHwQ) - [Inclusion-Exclusion](https://lightmysky.com/learn/mathematics/inclusion-exclusion-mt_MgU-IMpVHA) ## Opens up - [Generating Functions](https://lightmysky.com/learn/mathematics/generating-functions-mt_WAX3O-HjRk)