# Solving Non-Linear Recurrence Relation?

• FeDeX_LaTeX
In summary, the conversation discusses how to solve non-linear recurrence relations and specifically how to transform them into homogeneous linear recurrence relations. The process involves introducing the operator R and manipulating the equations to eliminate arbitrary constants.
FeDeX_LaTeX
Gold Member
Hello;

I do not have any experience in solving non-linear recurrence relations, so I was just wondering how one solves them.

For example, consider the sequence: 1, 2, 6, 15, 31, 56

In general, $$F_{n+1} = F_{n} + n^{2}$$

Do I still substitute $$F_{n} = k^{n}$$?

Thanks.

First transform into a homogeneous linear recurrence relation. In your case, a(n) = 4a(n-1) - 6a(n-2) + 4a(n-3) - a(n-4).

Hello;

Thanks for the reply. How did you transform it into a homogeneous linear recurrence relation? Did you use trial and error, or is there a method to do this (or is there something obvious I'm missing here)? Can all non-linear recurrence relations be transformed into homogeneous linear recurrence relations?

FeDeX_LaTeX said:
Hello;

I do not have any experience in solving non-linear recurrence relations, so I was just wondering how one solves them.

For example, consider the sequence: 1, 2, 6, 15, 31, 56

In general, $$F_{n+1} = F_{n} + n^{2}$$

Do I still substitute $$F_{n} = k^{n}$$?

Thanks.

Introduce the operator R that retards sequences:

$$R(f_n)=f_{n-1}$$

then you can express you equation as

$$(R-1)f_n=(n-1)^2$$

Multiply this by (R-1)^3 from the left, you'll get

$$(R-1)^4f_n=0$$

(check it...)

This last equation is linear and is of fourth order, the solutions depends on 4 arbitrary constants. 3 of then can be eliminated by substitution into the original equation.

Hello,

Thank you for your question. Solving non-linear recurrence relations can be a challenging task, but there are methods and techniques that can help us find the solution. One approach is to use the characteristic equation method, where we transform the recurrence relation into a polynomial equation and solve for its roots. Another method is to use generating functions, which can help us find a closed-form expression for the sequence. As for your specific example, substituting F_{n} = k^{n} may not always work, but it is a good starting point. I recommend doing some research and practicing with different examples to get a better understanding of how to solve non-linear recurrence relations. Best of luck!

## 1. What is a non-linear recurrence relation?

A non-linear recurrence relation is a mathematical equation that describes the relationship between a term in a sequence and one or more previous terms in that sequence. Unlike linear recurrence relations, which have a constant rate of change, non-linear recurrence relations have a varying rate of change.

## 2. Why is it important to solve non-linear recurrence relations?

Non-linear recurrence relations arise in many real-world problems, such as population growth, economic models, and physical systems. Solving these equations allows us to understand and predict the behavior of these systems, making them crucial in fields such as physics, biology, and economics.

## 3. What methods can be used to solve non-linear recurrence relations?

There are several methods for solving non-linear recurrence relations, including substitution, iteration, and generating functions. Each method has its advantages and is suitable for different types of equations. In some cases, advanced techniques such as matrix methods or calculus may also be used.

## 4. What are the challenges in solving non-linear recurrence relations?

Non-linear recurrence relations can be challenging to solve because there is no one-size-fits-all method. Each equation may require a different approach, and it may take trial and error to find the most suitable method. Additionally, the complexity of the equation may increase as the number of terms in the sequence grows, making it more difficult to find a solution.

## 5. How can solving non-linear recurrence relations benefit society?

Solving non-linear recurrence relations can have a significant impact on society by helping us understand and make predictions about complex systems. This knowledge can be used to make informed decisions in areas such as healthcare, economics, and technology, leading to advancements and improvements in our daily lives.

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