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Recursion Sequences - Concept

Teacher/Instructor Carl Horowitz
Carl Horowitz

University of Michigan
Runs his own tutoring company

Carl taught upper-level math in several schools and currently runs his own tutoring company. He bets that no one can beat his love for intensive outdoor activities!

While arithmetic and geometric sequences involve a rule that uses a constant number, recursion sequences use the terms themselves in the rule. One term in recursion sequences is determined from using the terms before it. This concept of recursion sequences can be difficult to fully comprehend, but is found often in mathematics. For example, the Fibonacci sequence is a famous recursion sequence.

So we have formulas for arithmetic sequences and geometric sequences. Okay? Whenever we are going from one term to the next by addition or subtraction or multiplication division, we know how to fit those 2 a formula. Okay? What we don't know is recursive formulas and what that means is that the term that we actually have depends on more than just the previous term and a set rate or difference, okay?
So what I have behind me is the fibonacci sequence. Okay? You may have seen it before but basically you start with 1 and 1 and then to get the next term, you always just to add the 2 terms before it. So to get the third term we add the first 2, 1+1 is 2. To get the next one, the 2 before it, before it. 1+2 is 3, 2+3 is 5, 3+5 the next one will be 8 so and so forth, okay? So we can't just write a term by the term right previous to it, okay? We have to actually have it written in terms of the terms before it and the one before that. So how we would actually write out the fibonacci sequence in sort of notation we're used to with our a sub ones and twos and n's is we define our first 2 terms as a sub 1 and then our ace of n is just going to be ace of n minus 2 which is actually 2 times before plus ace of n minus 1 which is the term before the term that we're looking at. It'a little hard to do from the abstract use so what I'm going to do is try one out, okay?
So what we have is 1, 2, 3, 4, 5. Let's try to find the sixth term. Ace of 6. To find the sixth term, we want to find ace of 6-2 just ace of 4 plus ace of 6-1, ace of 5. Ace of 4 is our fourth term, so we just have 1, 2, 3, 4. Ace of 5 is our next term which is going to be 5, and so ace of 6 is just going to be equal to 8, okay? So by just adding the 2 terms before it which we can write as ace of n-1 and minus 2 and together we are able to find terms in the fibonacci sequence. This is convenient for going down the row but it's not convenient for finding terms way down the line. If I want to find the hundredth term, I have to have the ninety ninth and the ninety eighth. In order to find the ninety ninth and the ninety eighth, I would need to know the ones before that as well. Okay?
So these aren't quite as easy to use as any other. Just ace of n is equal to a set formula, because in order to find any term we need to know the ones previous to it. Okay? But either way it's a good way to find a sequence when you're starting at the beginning and just moving forward down the line.

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