Extending geometric sequences | Mathematics I | High School Math | Khan Academy

So we're told that the first four terms of a geometric sequence are given. They give us the first four terms. They say, what is the fifth term in the sequence?

And like always, pause the video and see if you can come up with the fifth term. Well, all we have to remind ourselves is for a geometric sequence, for a geometric sequence, each successive term is the previous term multiplied by some number, and that number we call the common ratio.

So let's think about it. To go from negative 1/32, that's the first term, to 1/8, what do we have to multiply by? What do we have to multiply by? Let's see, we're going to multiply. It's going to be multiplied by a negative since we went from a negative to a positive. So we're going to multiply. We're going to multiply by negative, and then it's going to be a 1 over—let's see—to go from a 32 to an 8. Actually, it's not going to be a 1 over; it's going to be—this is 4 times as large as that. It's going to be negative 4.

Negative 1/32 times negative 4 is positive 1/8. Just to make that clear, negative 1/32 times negative 4. That's the same thing as times negative 4 over 1. It's going to be positive—negative times a negative is a positive—positive 4 over 32, which is equal to 1/8.

And let's see if that holds up. So to go from 1/8 to negative 1/2, we once again would multiply by negative 4. Negative 4 times 1/8 is negative 4/8, which is negative 1/2.

And so then we multiply by negative 4 again. So let me make it clear. We're multiplying by negative 4 each time. You multiply by negative 4 again, you get to positive 2. Because negative 4—negative negative 4 over negative 2—you can view it that way—is positive 2.

And so to get the fifth term in the sequence, we would multiply by negative 4 again. And so 2 times negative 4 is negative 8.

Negative 4 is the common ratio for this geometric sequence. But just to answer the question, what is the fifth term? It is going to be negative 8.