Introduction to geometric sequences
- [Voiceover] The geometric sequence a sub i, is defined by the formula where the first term, a sub one is equal to negative 1/8, and then every term after that is defined as being, so a sub i is going to be two times the term before that. A sub i is two times a sub i, minus one. What is a sub four, the fourth term in the sequence? Pause the video, and see if you can work this out. Well there's a couple of ways that you could tackle this. One is to just directly use these formulas. We could say that a sub four, well that's going to be this case right over here. A sub four is going to be equal to two times a sub three. Well, a sub three, if we go and use this formula, is going to be equal to two times a sub two. Each term is equal to two times the term before it. Then we go back to this formula again, and say a sub two is going to be two times a sub one. Two times a sub one. Lucky for us, we know that a sub one is negative 1/8. It's going to be two times negative 1/8, which is equal to negative 1/4. Negative 1/4. So this is negative 1/4. Two times negative 1/4, is equal to negative 2/4, or negative 1/2. A sub four is two times a sub three. A sub three is negative 1/2. So this is going to be two times negative 1/2, which is going to be equal to negative one. So that's one way to solve it. Another way to think about it is, look, we have our initial term. We also know our common ratio. We know each successive term is two times the term before it. So we could explicitly, this is a recursive definition for our geometric series. We could explicitly write it as a sub i is going to be equal to our initial term negative 1/8. Then we're going to multiply it by two. We're going to multiply it by two, i minus one times. So we could say times two, to the i minus oneth power. Let's make sure that makes sense. A sub one, based on this formula, a sub one would be negative 1/8, times two to the one minus one. Two to the zeroeth power. So that makes sense. That would be negative 1/8. Based on this formula, a sub two would be negative 1/8, times two to the two minus one. So two to the first power. We're going to take our initial term, and multiply it by two, once. Which is exactly right. A sub two is negative 1/4. So we want to find the fourth term in the sequence, we could just say well, using this explicit formula, we could say a sub four, is equal to negative 1/8, times two to the four, minus one. Two to the four, minus one power. So this is equal to negative 1/8, times two to the third power. This is negative 1/8 times eight. Negative 1/8 times eight, which is equal to negative one. You might be a little bit, a toss up on which method you want to use, but for sure this second method, right over here where we'd come up with an explicit formula once we know the initial term, and we know the common ratio, this would be way easier, if you were trying to find say, the 40th term. Because doing the 40th term recurs to be like this, would take a lot of time, and frankly, a lot of paper.