Graphing arithmetic sequences | Algebra I (TX TEKS) | Khan Academy

We are told that F of n is equal to F of n minus 1 plus 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 of one is equal to 8.

Whenever 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 equal 1, 2, 3, and 4, and then we're going to graph that and we're going to discuss that graph.

All right, now let's work through this together. So, let me in this column let we have n, and here I will have F of n. So, we'll start with n equals 1. That's pretty straightforward; they tell us that F of one is equal to 8. That was pretty straightforward. Now, let's go to when n equals 2. Well, F of two is equal to F of 2 minus 1, so it's equal to F of 1 plus 6.

Well, we know that F of one we just figured out is 8, so it's equal to 8 plus 6, which is equal to 14. Let's keep going, maybe in purple. All right, so now we want to figure out what F of 3 is going to be equal to. Well, same idea; it's going to be equal to F of 3 minus one or F of 2 plus 6. We keep adding six every time.

So, F of two we just figured out is 14. This is strangely fun! 14 plus 6, that is equal to 20. And then last but not least, maybe in light blue when n equals 4. Well, let's figure out F of four; it's going to be equal to F of three plus 6, which is equal to 20. F of 3 is 20 plus 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 8, 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. And 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 is equal to F of N. So, let's take that first point when n equals 1; the value of our function is 8. It gets you right about there. Then when n is 2, we get to 14. 2, 14, right about there.

When n is 3, we get to 20, so that is there. And then, last but not least, when n is 4, we get to 26. 26 gets us right about there. So, you might notice something very interesting here; it looks like these dots are on a line.

Now, this isn't a line because we're only defining this for whole number n's, 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.

So, if this were a line, if I were to try to connect these dots with a line, that line would have a slope of six because 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. If 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.