Recurrence Relation Solver

A recurrence relation is a mathematical equation that defines a sequence recursively. Each term of the sequence is expressed as a function of the preceding terms. Linear homogeneous recurrence relations with constant coefficients are among the most common types studied in discrete mathematics and computer science.

This calculator solves second-order linear homogeneous recurrence relations of the form:

$$a \cdot a_{n+2} + b \cdot a_{n+1} + c \cdot a_n = 0$$

where $a$, $b$, and $c$ are constants, and the initial conditions $a_1$ and $a_2$ are given.

1. Enter the coefficients: Input the values for $a$, $b$, and $c$ from your recurrence relation equation.
2. Enter initial conditions: Provide the values for $a_1$ and $a_2$ (the first two terms of your sequence).
3. Click "Solve": The calculator will compute the closed-form solution using the characteristic equation method.

Example: Solve the recurrence relation $a_{n+2} - 5a_{n+1} + 6a_n = 0$ with initial conditions $a_1 = 1$ and $a_2 = 5$.

Solution:

1. The characteristic equation is: $r^2 - 5r + 6 = 0$
2. Factoring: $(r - 2)(r - 3) = 0$, so $r = 2$ and $r = 3$
3. Since we have two distinct real roots, the general solution is: $$a_n = A \cdot 2^{n-1} + B \cdot 3^{n-1}$$
4. Using the initial conditions, we find $A = -2$ and $B = 3$
5. Therefore: $$a_n = -2 \cdot 2^{n-1} + 3 \cdot 3^{n-1}$$

Try entering $a = 1$, $b = -5$, $c = 6$, $a_1 = 1$, and $a_2 = 5$ into the calculator to verify this result!

The solution method used by this calculator is based on the characteristic equation technique:

Step 1: Characteristic Equation

For the recurrence relation $a \cdot a_{n+2} + b \cdot a_{n+1} + c \cdot a_n = 0$, we form the characteristic equation:

$$a \cdot r^2 + b \cdot r + c = 0$$

Step 2: Find Roots

We solve for $r$ using the quadratic formula:

$$r = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$$

Step 3: General Solution

The form of the solution depends on the discriminant $D = b^2 - 4ac$:

• $D > 0$ (Distinct real roots $r_1$, $r_2$): $$a_n = A \cdot r_1^{n-1} + B \cdot r_2^{n-1}$$
• $D = 0$ (Repeated root $r$): $$a_n = (A \cdot (n-1) + B) \cdot r^{n-1}$$
• $D < 0$ (Complex roots):
  Currently not supported by this calculator

Step 4: Determine Constants

The constants $A$ and $B$ are determined by substituting the initial conditions $a_1$ and $a_2$ into the general solution.

Recurrence relations have numerous applications in various fields:

• Computer Science: Analyzing algorithm complexity, especially for recursive algorithms like merge sort and binary search
• Mathematics: Studying sequences like Fibonacci numbers, Lucas numbers, and other mathematical sequences
• Physics: Modeling physical systems with discrete time steps
• Economics: Modeling population growth, financial calculations, and economic forecasting
• Biology: Population dynamics and genetic modeling