# Solution of a recursion relation

• ledamage
In summary, The speaker is wondering if there is an explicit solution to a recursion relation of the form αn+2 = A αn+1 + B αn + Cn, and mentions that the solution without Cn can be easily computed. They also mention not finding anything on the internet.

#### ledamage

Hi there!

I wonder if there's an explicit solution to a recursion relation of the form

$$\alpha_{n+2} = A \alpha_{n+1} + B \alpha_n + C^n$$ .

The solution of this recursion relation without $C^n$ can easily be computed. I haven't found anything on the net.

Thanks!

ledamage said:
Hi there!

I wonder if there's an explicit solution to a recursion relation of the form

αn+2 = A αn+1 + B αn + Cn .

The solution of this recursion relation without Cn can easily be computed. I haven't found anything on the net.

Thanks!

Hi ledamage!

You need to find a particular solution, exactly as if this was a differential equation.

## 1. What is a recursion relation?

A recursion relation is a mathematical equation or formula that defines a sequence of values in terms of one or more previous values in the sequence. It is a type of mathematical induction where the solution of the equation depends on previous terms in the sequence.

## 2. How is a recursion relation solved?

A recursion relation can be solved by breaking it down into smaller sub-problems and solving them recursively. This means that the solution for a particular term in the sequence depends on the solution for previous terms, until a base case is reached.

## 3. What is a base case in a recursion relation?

A base case in a recursion relation is the starting point or initial condition of the sequence. It is the value that does not depend on any previous terms and is used to terminate the recursive process.

## 4. What are the advantages of using recursion relations?

Recursion relations are useful in solving complex problems that can be broken down into smaller, simpler sub-problems. They can also lead to more elegant and concise solutions compared to iterative methods. Additionally, some problems are more naturally expressed and solved using recursion.

## 5. What are the limitations of recursion relations?

One limitation of recursion relations is that they can be computationally expensive, as each recursive call requires additional memory and processing time. Additionally, if the base case is not properly defined, it can lead to infinite recursion and cause the program to crash. It also requires a good understanding of the problem and proper implementation to avoid errors and inefficiencies.