Sequences

Daniel123 · 2007-09-06 04:33:24

Hello

I just started sequences today, and I just wanted to check my answer for this. It seems right, but I'm new to it so better play safe .

Suggest a possible recurrence relationship for the following sequence:

4, 10, 18, 38, 74, ...

I have another quick question:

Why are recurrence relationships written in the form, for example:

?

Thanks.

Prakash Panneer · 2007-09-06 05:00:15

The reason is...

Sequence is the arrangement of numbers which follows a pattern.

The rule behind the arrangement of numbers is called Pattern.

That is, the relation between the those number.

Hope you have understood this one.

krassi_holmz · 2007-09-06 05:14:19

Daniel123 wrote:

Why are recurrence relationships written in the form, for example:
?
Thanks.

That's not true. You can write it as you like. The difference is only in the domain for n-s. For example, if you're specifying the Fibonnaci sequence, you may write:

or:

Daniel123 · 2007-09-06 05:32:17

Oh ok, its just that in my book it alwys has the

version. By the way, no one said if my answer above was correct!

mathsyperson · 2007-09-06 06:05:29

Your answer looks perfect. I'd move things around a bit and write it as:

They're both the same thing though.

Daniel123 · 2007-09-06 06:13:43

Thanks mathsy

Last edited by Daniel123 (2007-09-06 06:13:52)

Daniel123 · 2007-09-06 09:16:14

I have another that I'm stuck on...

If you give me an answer can you please tell me your thought processes? I need to train my brain to do these, as there seems no specific method for the more complicated ones..

Find the recurrence formula that defines:

I can see that the numerator is increasing by 1 and the denominator by 2.. but how would I express that algebraically?

Thanks.

luca-deltodesco · 2007-09-06 09:25:51

its very easy to write that as an n'th formula:

it seems rather pointless writing it as a recurrence equation when you can so easily do it as a full equation, but i guess you could write it like this:

bossk171 · 2007-09-06 09:26:12

How about:

My thought process:

all even numbers can be expressed as 2n and all odd numbers are (2n-1) or (2n+1). The top is simply n, the bottom is aways odd. 2n+1 doesn't work, no how about 2n-1.

bossk171 · 2007-09-06 09:27:19

I'm just too slow...

Daniel123 · 2007-09-06 09:53:34

Thanks..

Ok I'm getting confused...

1. what is the difference between an n'th formula and a recurrence formula?
2. when do you use each of them?

mathsyperson · 2007-09-06 21:45:40

An nth term formula uses where a term is in the sequence to determine what the term is.
A recurrence relation uses the terms before to determine it.

For example, the 7th term in a sequence would be determined entering the number 7 into some function if using the nth term.
But using recurrence relations, the 6th (and maybe 5th or even 4th) terms would be used instead.

Your previous example was actually a mix of the two, because it involved U_(n-1) and also n.

As for when to use each of them, it really just depends on which is easier.
The 1, 2/3, 3/5, 4/7 sequence that you posted is very easy to find an nth term for, but a recurrence relation is trickier.

Conversely, something like the Fibonacci sequence [U_n = U_(n-1) + U_(n-2)] is easy to define a recurrence relation for, but the nth term is more complicated.

If it's equally easy to use either, I would recommend using the nth term, because it's far easier to find, for example, the 100th term using that.

Daniel123 · 2007-09-07 02:57:00

I understand now. So is there no clear cut method for finding the formula that defines a sequence? It's just a matter of spotting the relationship between a term and its position, or a term and its previous term?

Thanks again!

Ricky · 2007-09-07 03:33:43

Not only is there no method of finding a formula or recurrence relation, but there are sequences that have none. Not only that, but there are more that have none than there are that have some sort of formula.

mikau · 2007-09-07 03:53:44

how can we verify that? aren't there an infinite number of formulas we can produce for a sequence? example, nth term = a*n^2 + bx + c, where a b c are real numbers.

mathsyperson · 2007-09-07 04:43:32

An nth term can always be found for finite sequences. If each term is random then the nth term will (probably) be very messy and involve n^(k-1), where there are k terms, but it will nonetheless exist.

Infinite sequences exist that can't be described by a formula though.

Ricky · 2007-09-07 05:11:10

mikau wrote:

how can we verify that? aren't there an infinite number of formulas we can produce for a sequence? example, nth term = a*n^2 + bx + c, where a b c are real numbers.

First, remember that we are talking about integer sequences. But we can only represent a countable amount of sequences, simply by the fact that all the knowledge we could possibly have is countable. To see this, just consider the sequence 0 1 00 01 10 11 000 001 010 011 010... Now interpret each of those numbers as ascii. Eventually, if we get high enough, Shakespeare's plays are all going to be in that list. Any idea any human can communicate is as well.

But the number of functions on the integers are uncountable. Can you prove it?

Math Is Fun Forum

#1 2007-09-06 04:33:24

Sequences

#2 2007-09-06 05:00:15

Re: Sequences

#3 2007-09-06 05:14:19

Re: Sequences

#4 2007-09-06 05:32:17

Re: Sequences

#5 2007-09-06 06:05:29

Re: Sequences

#6 2007-09-06 06:13:43

Re: Sequences

#7 2007-09-06 09:16:14

Re: Sequences

#8 2007-09-06 09:25:51

Re: Sequences

#9 2007-09-06 09:26:12

Re: Sequences

#10 2007-09-06 09:27:19

Re: Sequences

#11 2007-09-06 09:53:34

Re: Sequences

#12 2007-09-06 21:45:40

Re: Sequences

#13 2007-09-07 02:57:00

Re: Sequences

#14 2007-09-07 03:33:43

Re: Sequences

#15 2007-09-07 03:53:44

Re: Sequences

#16 2007-09-07 04:43:32

Re: Sequences

#17 2007-09-07 05:11:10

Re: Sequences

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