When k is 2, we get 4. So this is another way of defining this infinite sequence. This is essentially a function, where an allowable input, the domain, is restricted to positive integers. If k is equal to 3, you get 3 times 2 plus 1 is 7. This is an explicit function. And so then we could write a sub k is equal to the previous term.

So a given term is equal to the previous term. And you keep adding 4. The sequences where you keep adding the same amount, we call these arithmetic sequences, which we will also explore in more detail. This right over here is a plus 3.

So we go to 3, to 7, to 11, So this is a sub k minus 1. Recursive and explicit sequences Video transcript What I want to do in this video is familiarize ourselves with the notion of a sequence.

And when you define a sequence recursively, you want to define what your first term is, with a sub 1 equaling 1. But I want to make us comfortable with how we can denote sequences and also how we can define them.

I want to make it clear-- I have essentially defined a function here. So how would we do this one?

And once again, you start at 3. When k is 3, we added 3 twice. And you can verify that this works. If I wanted a more traditional function notation, I could have written a of k, where k is the term that I care about.

So both of these, this right over here is a recursive definition. Well, what about a sub 3? Now, I could also define it by not explicitly writing the sequence like this.

Now, how would I denote this business right over here? And I want to be clear-- not every sequence can be defined as either an explicit function like this, or as a recursive function. So whatever k is, we started at 1. For the fourth term, we add 4 three times. You can define every other term in terms of the term before it.

When k is 3, we get 7.

So this right over here is explicit. And this right over here would be the first term. So this one we would call a finite sequence. Well, I could say that this is equal to-- and people tend to use a.

You get to 7. This right over here was a plus 3. Well, once again, we could write this as a sub k. Starting at k, the first term, going to infinity with-- our first term, a sub 1, is going to be 3, now. Let me write this in. I could also have an infinite sequence.

Now, how does this make sense? We started with kind of a base case. But many can, including this, which is an arithmetic sequence, where we keep adding the same quantity over and over again. But this is all they refer to.A recursive definition of a function defines values of the functions for some inputs in terms of the values of the same function for other inputs.

For example, the factorial function n! is defined by the rules 0! = 1. (n+1)! = (n+1)·n!. This definition is valid for all n, because the recursion eventually reaches the base case of 0. Recursive definition, pertaining to or using a rule or procedure that can be applied repeatedly.

See more. The key to recursive writing is recognizing that writing is a process that repeats. Do not think of writing as five neat steps that lead to completion, and then you never visit the paper again.

Rather, think of writing as developing, stopping, sharing, and changing. Recursive definition is - of, relating to, or involving recursion.

How to use recursive in a sentence. of, relating to, or involving recursion; of, relating to, or constituting a procedure that can repeat itself indefinitely.

Recursive Definitions A recursive formula always uses the preceding term to define the next term of the sequence. Sequences can have the same formula but because they start with a different number, they are different patterns.

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