How to solve recurrence relation with one real root and two complex roots ?

To solve the recurrence relation with one real root and two complex roots, you can start by rewriting the relation as a_n = -a_(n-2). Then, you can find the roots of the equation r^3 - r^2 - 1 = 0 and use them to find a_n by plugging them into the equation. The pattern for a_n is a_4 = a_2 = -1 and a_5 = a_3 = 1. This simplifies the process and helps you find the solution faster."In summary, to solve a recurrence relation with one real root and two complex roots, you can rewrite the relation and find the roots of the equation. Then, use these roots to
  • #1
huifei
1
0
How to solve recurrence relation with one real root and two complex roots ?

The Example is ;

Solve the recurrence relation a n-1 + a n-3 = 0 where n ≥ 3 and a 1 = 1 a 2 = 1 a 3 = 2
a n = nth order
a n-1 = (n-1)th order.
a n-3 = (n-3)th order.

I've started the solving ;
a n = r^n
so
the equation will be ;

r^n - r^(n-1) - r^(n-3)
r^3 - r^2 - 1 = 0
I could'nt do anything after find the roots ?
What should i do ?
Thanks for helping.
 
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  • #2


I think you are making this harder than it is. Hint:

you can rewrite you recurrence relation as

[tex]
a_{n} = - a_{n-2}.
[/tex]

So what is [tex]a_4[/tex]? How about [tex]a_5[/tex]? See the pattern?

jason
 

Related to How to solve recurrence relation with one real root and two complex roots ?

1. How do I identify the type of roots in a recurrence relation?

This can be done by finding the characteristic equation of the recurrence relation and solving for its roots. If the roots are real and distinct, the recurrence relation will have a solution in the form of a linear combination of the roots raised to the power of the index. If the roots are complex, the solution will involve complex exponentials.

2. What is the process for solving a recurrence relation with one real root and two complex roots?

The first step is to solve for the real root using the method for solving linear recurrence relations. Once the real root is found, the complex roots can be found using the quadratic formula. Then, the solution can be expressed as a linear combination of the real root and the complex roots raised to the power of the index.

3. Can a recurrence relation with one real root and two complex roots have a closed-form solution?

Yes, it is possible for a recurrence relation with one real root and two complex roots to have a closed-form solution. However, the solution will involve complex numbers and may not be easily expressed in a simple form.

4. How do I handle the complex roots when solving a recurrence relation?

When solving a recurrence relation with complex roots, it is important to remember that complex numbers have both a real and imaginary part. The solution will involve using both the real and imaginary parts in the final expression. It may also be helpful to convert the complex numbers to polar form for easier manipulation.

5. What are some common pitfalls when solving a recurrence relation with one real root and two complex roots?

One common mistake is forgetting to include the real and imaginary parts of the complex roots in the final solution. Another mistake is not using the correct formula for converting complex numbers to polar form. It is also important to carefully check the algebraic steps when simplifying the solution to avoid any errors.

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