Graphing geometric sequences | Algebra 1 (TX TEKS) | Khan Academy

We're told a sequence is defined by F of n is equal to 1/5 * F of n-1. So each term, whatever the value of the function is there, where the sequence is for that term, it's 1 times the previous term for each whole number n, where n is greater than one.

Then they also tell us which we need: what the first term is; F of one is equal to 50. Now, the reason I have this graph here is what I want you to do is pause this video and figure out what the value of this function is for n = 1, 2, 3, and 4. Then we're going to be able to graph this together and think about what that graph looks like and why.

All right, now let's do this together. What I like to do is set up a table here, where on this left column I have n, and then over here I have F of n on the right column.

Let's first start with n equals 1. Well, they already tell us that F of one is equal to 50. If we want, we can plot that when n is equal to one. I should say this is the n axis instead of the x axis. When n is equal to 1, F of one is 50 right over here. So let's call this the Y is equal to F of n axis.

All right, let's do the next one. I'll do that in red. When n is equal to 2, well, F of two is going to be equal to it says it right over here: 1/5 times F of (2 - 1) or 1 * F of 1. So I could just write that as 1 * F of 1, which is equal to 1/5 * 50, which is equal to 10.

So when n is equal to two, this is equal to 10 right over here; F of two is 10. Now let's go to when n is equal to 3. F of three, I think you see the pattern here, is equal to 1/5 times F of 2. We know what F of two is; it's 10, so it's equal to 1/5 * 10, which is equal to 2.

So when n is equal to 3, Y is equal to F of n is equal to 2, which is right about there. And then, last but not least, in orange, when n is equal to 4, F of 4 is equal to 1/5 times F of three, the previous term, which is equal to 1/5 * 2, which is equal to two-fifths.

So that's less than one, so it's going to be real just right above zero like that. We have graphed those four points, and you might see an interesting pattern here. You might say, "Hey, you know what? This looks a lot like exponential decay."

That's not a coincidence because remember, every term here it's 1/5 times the previous term. So we're decaying; we're multiplying each successive term by 1/5; it's getting smaller and smaller and smaller. But that's what we're seeing here when we're dealing with a geometric sequence.

Now, in this particular scenario, we defined this geometric sequence recursively. Each successive term we've defined in terms of the previous term, and then we got a starting condition. There's other ways to define a geometric series so that it is not recursive, but it's good to get exposure to this.

Generally speaking, if you have each successive term, it's going to be some multiple of the previous term. Here, it's a multiple less than one; it could be a multiple greater than one. You're going to have points that look like they're on some type of exponential curve.

If, on the other hand, you had an arithmetic sequence, where each successive term is plus or minus some fixed amount of the previous term, then it will look more linear.