Find closed formula for the sequence.

In summary, the conversation discusses finding a closed formula for the sequence 0,1,3,0,1,3,0,1,3... and suggests using techniques such as the "mod" function or complex powers. The use of a cube root of -1 is also mentioned.
  • #1
jjangub
24
0
I have a problem with this question.
The question is "Find closed formula for the sequence 0,1,3,0,1,3,0,1,3..."
It can be written as (0,1,0,0,1,0,0,1,0...)+3*(0,0,1,0,0,1,0,0,1...)
I know the sequence of 0,0,1,0,0,1..., but I do not know how to get the
sequence of 0,1,0,0,1,0...
And is there any easier way to get the sequence of 0,0,1,0,0,1...or 0,1,0,0,1,0...?
Thank you...
 
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  • #2
jjangub said:
I have a problem with this question.
The question is "Find closed formula for the sequence 0,1,3,0,1,3,0,1,3..."
It can be written as (0,1,0,0,1,0,0,1,0...)+3*(0,0,1,0,0,1,0,0,1...)
I know the sequence of 0,0,1,0,0,1..., but I do not know how to get the
sequence of 0,1,0,0,1,0...
And is there any easier way to get the sequence of 0,0,1,0,0,1...or 0,1,0,0,1,0...?
Thank you...

Depends on what you're allowed to use, and what techniques you know. In Pari it would be
Code:
[3,0,1][n%3+1]
. The "mod" function would make quick work of it. Complex powers could work: with w a cube root of -1, [tex]w^k+\bar{w^k}[/tex] is 1, -1, or 2 depending on k mod 3. Etc.

Edit: The conjugate symbol isn't showing up very well...
 

1. What is a closed formula for a sequence?

A closed formula, also known as a closed form or explicit formula, is a mathematical expression that can be used to calculate any term of a sequence without the need to know the previous terms.

2. How do you find a closed formula for a sequence?

To find a closed formula for a sequence, you will need to identify the pattern or rule that governs the sequence. This can be done by examining the first few terms and looking for a relationship between them. Once the pattern is identified, you can use it to write a general formula that can be used to find any term in the sequence.

3. Why is it important to find a closed formula for a sequence?

Finding a closed formula for a sequence can help us understand the behavior and properties of the sequence. It also allows us to calculate any term in the sequence without having to manually calculate each term one by one. Additionally, closed formulas can be used for various applications in fields such as physics, chemistry, and engineering.

4. Can all sequences have a closed formula?

No, not all sequences have a closed formula. Some sequences have a random or chaotic pattern, making it difficult to find a general formula. Other sequences may have a pattern that is too complex to be expressed in a simple closed form.

5. Are there different methods for finding a closed formula for a sequence?

Yes, there are different methods for finding a closed formula for a sequence, such as the method of common differences, method of finite differences, and method of undetermined coefficients. These methods may vary in complexity and effectiveness depending on the sequence and its pattern.

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