Graphing arithmetic sequences

0:00- [Tutor] We are told that f(n) = f(n-1) +6. So the value of this function for each term N is defined in terms of the value of the function for previous terms, we're essentially adding six to the previous term, for each whole number N, where N is greater than one and f(1) = 8.

0:22Whenever you define something recursively like this, where you're defining it in terms of a previous term, you have to set up an initial point that you can start with and we'll see in a second why that's important. Now what I want you to do is pause this video and based on this definition, figure out what the value of the function is for N = 1, 2, 3, and four.

0:43And then we're going to graph that and we're going to discuss that graph. All right, now let's work through this together. And so let me, in this column, let me have N, and here I will have f(n). So we'll start with N = 1. That's pretty straightforward. They tell us that f(1) = 8.

1:04That was pretty straightforward. Now let's go to when N = 2. Well, f(2) is = f(2-1). So it's = f(1) + 6. Well, we know that f(1), we just figured it out is eight, so it's equal to eight plus six, which is equal to 14. Let's keep going, maybe in purple. All right, so now we want to figure out what f(3) is going to be equal to.

1:37Well, same idea. It's going to be equal to f(3) -1 or f(2) + 6. We keep adding six every time. So f(2) we just figured out is 14. This is strangely fun. 14 plus six, that is equal to 20. And then last but not least, maybe in light blue, when N = 4, well let's figure out.

2:04f(4) is going to be equal to f(3) + 6, which is equal to 20. f(3) is 20 + 6, which is equal to 26. So you might have noticed a pattern here. We start with when on our first term, the value of the function is eight. And then what did we do? We added six. And then to get to the next term, we added six again, and then we added six again.

2:31And so we should see that visually when we actually try to graph it. So let's graph it here. And actually, instead of calling this the X axis, let me call this the N axis, and the Y axis, let's just call that Y = f(n). And so let's take that first point when N = 1, the value of our function is 8.

2:511, 8. It gets you right about there. Then when N is two, we get to 14. 2 14, right about there. When N is three, we get to 20. So that is there. And then last but not least, when N is four, we get to 26. So 26 gets, its right about there. So you might notice something very interesting here.

3:17It looks like these dots are on a line. Now this isn't a line 'cause we're only defining this for a whole number ends, but we can see it looks like a line. And every time we move forward by one, we are moving up by six. We move forward by one, we're moving up by six.

3:38So if this were a line, if I were to try to connect a dot or connect to these dots with a line, that line would have a slope of six 'cause our change in N is one, and then our change in Y or change in the value of our function is going to be six every time. So in general, if someone shows you a sequence like this, and this is really an arithmetic sequence where each term is a previous term, plus or minus some fixed amount, you're going to see something that looks linear.

4:06If you saw a curve then that wouldn't, or something like dots on a curve, then that wouldn't be an arithmetic sequence, that would be something else. But if you see dots that seem to form or be points on a line, that's a pretty good clue that you're dealing with an arithmetic sequence.