# Reduction of Order For Recursions

• Insights
• topsquark
In summary, recursion relations, also known as recurrence relations, are a mathematical concept commonly taught in secondary school. They involve starting with a number and repeatedly adding a constant value to it to create a series of numbers. This process can be represented by a rule, such as ##a_{n + 1} = a_n + 3##, where ##a_0 = 1##. Recursion relations can be used to represent more complex series, such as the Fibonacci series, and can also be solved using finite difference calculus.

#### topsquark

Gold Member
MHB
This is not meant as a full introduction to recursion relations but it should suffice for just about any level of the student.
Most of us remember recursion relations from secondary school. We start with a number, say, 1. Then we add 3. That gives us 4. Now we that number and add 3 again and get 7. And so on. This process creates a series of numbers ##\{ 1, 4, 7, 10, \dots \}##. Once we get beyond that stage we start talking about how to represent these. Well, let’s call the starting number ##a_0##. Then the next number in the series would be ##a_1##, then ##a_2##, …, on to ##a_n## where n is the nth number in the series. We may now express our recursion as a rule: ##a_{n +1} = a_n + 3##, where ##a_0 = 1##.
We may go much further. We can talk about things like the Fibonacci series: ##F_{n + 2} = F_{n + 1} + F_{n}##, where ##F_1 = F_2 = 1##. This is the famous series ##\{ 1, 1, 2, 3, 5, 8, 13, \dots \}##. But what if we want a formula to find out what ##F_n## is for the nth number...

malawi_glenn, pbuk, fresh_42 and 1 other person
topsquark said:
Is the term "recursion relation" in common use? I would call it a "recurrence relation".

Also I would say "finite difference calculus" instead of "difference calculus".

topsquark
pbuk said:
Is the term "recursion relation" in common use? I would call it a "recurrence relation".

Also I would say "finite difference calculus" instead of "difference calculus".
Sorry! As I understand it both terms (the recursive and finite difference) are in use. However, the only work I've done with recurrence relations comes from High School and I couldn't really find much on the web about finite difference calculus. I had to fill in a lot of gaps to do this, so I may not be using standard terminology.

-Dan

topsquark
pbuk said:
I knew most of the article, but there were a number of topics in the links that are new to me. Thanks!

-Dan

jim mcnamara