Home

Characteristic Equation Method Calculator

Online second-order recurrence calculator: given a_n=p·a_(n−1)+q·a_(n−2), iterate to the nth term and show the characteristic roots; Fibonacci is the classic case.

展开更多 ▾
a_n=p·a_(n−1)+q·a_(n−2); r²−pr−q=0
First term
Second term
Coefficient
Coefficient
Positive integer

📖 Tutorial | Characteristic Eq

1. Definition
The second-order recurrence a_n=p·a_(n−1)+q·a_(n−2) has the characteristic equation r²−pr−q=0 with roots (p±√(p²+4q))/2.
2. Formula & Notation
a₁,a₂Two initial values
3. Properties & Laws
  • Characteristic equation r²−pr−q=0;
  • Distinct roots give A r₁^n+B r₂^n;
  • Fibonacci has p=q=1 and the golden ratio;
  • Repeated/complex roots differ.
4. Steps
  1. Enter a₁,a₂,p,q,n;
  2. Click "Find Term" for roots and a preview;
  3. Or click "Load Example".
5. Worked Examples
Example: Fibonacci 1,1,2,3,5 gives term 10 = 55.
6. Common Pitfalls
Needs two initial values;
The constant term is −q;
Complex roots still give real terms.

❓ FAQ | Characteristic Eq

Equation?
r²−pr−q=0.
Roots?
(p±√(p²+4q))/2.
Fibonacci?
p=q=1.
Golden ratio?
≈1.618.
Initial values?
Two.
Negative discriminant?
Complex conjugates, real terms.
Closed form?
Linear combo of roots.