Questions about using iteration to express a recurisve sequence

  • Thread starter mr_coffee
  • Start date
  • Tags
    Sequence
In summary, the conversation discusses finding an explicit formula for a recursively defined sequence using iteration. The method of guessing a formula is mentioned, but the individual is unsure of how to make an educated guess. Another mistake is pointed out in their work, and a formula is suggested involving a geometric series. The individual also questions how to determine the exponents in the formula and suggests using a formula to find the sum of the series.
  • #1
mr_coffee
1,629
1
Hello everyone, I'm having some issues on a few problems and I'm not sure if i did the first one I'm going to post right, any help or clarifcation would be great.

The directions are: In eavch of 3-15 a sequence is defined recursively. Use iteration to guess an explicit formula for the sequence.

and here is my work:
http://img518.imageshack.us/img518/8013/scan0001pa5.jpg



For the last one, I see that the 2 is increasing at the same rate as the 3^k, but i have no idea about the pattern, I'm not sure what technique I'm suppose to use, i know they said "guess" a formula, but what will help me make an educated guess? Of course there are some easy ones like n! that i can recongize but these arn't so easy for me to see what's going on.

I can see i can factor out a 2 and a 3^2 from p_4 but i don't see how that is going to help me much, getting:
[tex]2^4 + 3^4 + 2*3^2(1 + 3)[/tex]

thanks!
 
Last edited by a moderator:
Physics news on Phys.org
  • #2
You made a bit of an arithmetic mistake: that should be [tex]2^2 3^2[/tex], not [tex]2^1 3^2[/tex]. If you continue on I think you'll notice that [tex]p_n = 2^n + 3^n + \left( 2^{n-2} 3^2 + 2^{n-3} 3^3 +...+ 2^1 3^{n-1} \right)[/tex]. That stuff in parentheses is a geometric series.
 
  • #3
Thanks durt!
I'm alittle confused on how you got some of the terms of n
[tex]p_n = 2^n + 3^n + \left( 2^{n-2} 3^2 + 2^{n-3} 3^3 +...+ 2^1 3^{n-1} \right)[/tex]

But the parts of it I don't understand is how do you know after looking at a few terms, like a_1 to a_4 how you figure out which exponents go to a certian power, such as [tex]2^{n-2} 3^2 + 2^{n-3} 3^3 +...+ 2^1 3^{n-1}[/tex] ?

I see that [tex]2^n + 3^n[/tex] would be just n because they are increasing at the same time, but from there what's the process to break that down into the other powers of n?

Thanks

Also I could replace that geometric series with the following sum right?
First term is [tex]2^{n-2} 3^2[/tex]
ratio is: 3/2 I believe because 2 is decreasing and 3 is increasing as you progress
term after the last term is: [tex]3/2* 2^1 3^{n-1}[/tex] = [tex]3^{n-2}[/tex]

so now applying a formula:
[mythical next term - first term]/(ratio-1)
= ( [tex]3^{n-2}[/tex] - [tex]2^{n-2} 3^2[/tex] )/(3/2 -1)
 
Last edited:

Related to Questions about using iteration to express a recurisve sequence

1. What is iteration?

Iteration is a process of repeatedly executing a set of instructions until a certain condition is met. It is commonly used in computer programming to solve problems that require repetitive actions.

2. How is iteration used to express a recursive sequence?

In a recursive sequence, the next term is defined in terms of the previous term. Iteration can be used to express this by repeatedly applying the same function or algorithm to the previous term to generate the next term.

3. What is the difference between recursion and iteration?

Recursion is a programming technique where a function calls itself until a base case is reached, while iteration is a process of repeating a set of instructions until a certain condition is met. In simple terms, recursion is a form of iteration, but it involves a function calling itself.

4. How do you know when to use iteration to express a recursive sequence?

You can use iteration to express a recursive sequence when the sequence has a well-defined pattern that can be expressed using a mathematical function or algorithm. In other words, if the next term in the sequence can be generated by repeatedly applying the same process to the previous term, then iteration can be used.

5. Can any recursive sequence be expressed using iteration?

No, not all recursive sequences can be expressed using iteration. Some recursive sequences may have complex patterns or rely on external factors, making it difficult to express them using a simple iterative process. In these cases, recursion may be a better approach.

Similar threads

  • Calculus and Beyond Homework Help
Replies
1
Views
450
  • Calculus and Beyond Homework Help
Replies
2
Views
294
  • Calculus and Beyond Homework Help
Replies
2
Views
529
  • Calculus and Beyond Homework Help
Replies
1
Views
760
  • Calculus and Beyond Homework Help
Replies
12
Views
1K
  • Calculus and Beyond Homework Help
Replies
9
Views
2K
  • Calculus and Beyond Homework Help
Replies
25
Views
599
  • Calculus and Beyond Homework Help
Replies
2
Views
478
  • Calculus and Beyond Homework Help
Replies
3
Views
1K
  • Calculus and Beyond Homework Help
Replies
2
Views
4K
Back
Top