# Sequences and Series: Solving recurrence systems

## Sequences and Series: Solving recurrence systems

## Understanding Recurrence Systems

- A
**recurrence system**is system of equations where the result of one equation is used as an input for the next. - Recurrence systems are often used in modelling sequences and series.
- The order of a recurrence system is the number of steps back it depends on. A system depending on the previous two steps is second order, and so on.
- The simplest form of a recurrence system is the first order linear recurrence relation:
`a(n) = pa(n-1) + q`

, where p and q are constants. - A higher order recurrence system could look like this:
`a(n) = pa(n-1) + qa(n-2) + r`

, where p, q, and r are constants, and this is a second order system.

## Solving Recurrence Systems

- Solving a recurrence system involves finding a general term
`a(n)`

that can calculate any term in the sequence without referring to previous terms. - The foundation of solving recurrence system lies in understanding the concept of a
**characteristic equation**. The characteristic equation of the recurrence relation`a(n) = pa(n-1) + qa(n-2)`

is`r^2 - pr - q = 0`

. - The roots of the characteristic equation provide valuable information about the solution to the recurrence system.
- When both roots are real and distinct,
`r1`

and`r2`

, the general form of the solution is`a(n) = kr1^n + lr2^n`

. - When the characteristic equation has repeated roots,
`r1`

, the general form of the solution is`a(n) = (kr + l)n^r`

. - When the roots are complex, arranged as
`p ± qi`

, the general solution is`a(n) = r^n[kcos(nθ) + lsin(nθ)]`

, where`r = sqrt(p^2 + q^2)`

and`θ = tan^-1(q/p)`

. - Constants
`k`

and`l`

can be found by using the initial conditions of the problem.

## Solving Recurrence Systems with Non-Homogeneous Terms

- In many realistic situations, the sequence is subjected to forces external to the system, leading to non-homogeneous recurrence relations.
- The procedure to solve such systems involves finding the
**complementary function**(solution of the homogeneous part) and a**particular solution**(solution of the non-homogeneous part). - We find the final solution by adding together the complementary function and the particular solution.

## Applications of Recurrence Systems

- Recurrence systems, due to their inherent iterative nature, are useful in modelling various practical phenomena including population growth, economic behaviours, and biological processes.
- Being able to solve recurrence systems opens up the ability to analyse complex dynamic systems in several fields including mathematics, physics, economics, and computer science.