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.
What a learner can do afterwards
- 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
1 · Read
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.
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.
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.
Write the rule, solve the characteristic equation, blend the root powers, and fit the blend to the start.
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Where it sits
8 questions wait behind this lesson, each with its answer explained. Every answer feeds the sky: stars light as they are learned, and dim when it is time to come back.