Welcome back, everyone. So when we introduced sequences, we saw equations for a_n involving n. The idea was that these are your inputs. You plug in 1, 2, 3 into the equations, and then you get the terms out of it. But sometimes that's going to be backward. You're going to have to look at a sequence of numbers or terms, and you're going to have to figure out what the formula for it is. And to do that, I'm going to show you in this video that we're going to be writing general formulas, or sometimes they're called explicit formulas. These are equations involving n where you plug in n's and you get the general terms out of it. What I'm going to show you here is that it really just comes down to finding the patterns that are going on between the numbers. So I'm going to show you a bunch of different examples of the most common types of patterns because when you see them, it means that your formula is going to contain some of these expressions. We're going to work a bunch of examples together. Let's go ahead and get started. Alright. So let's take a look at the first pattern here. So oftentimes in your sequences, you'll see numbers that increase by the same amount each time. So for example, 5, 6, 7, 8, 9. If you look at these, there's a pattern here. All the numbers increase by 1 each time. And we've also seen other examples where numbers increase by 2, like 2, 4, 6, 8, 10, or 3, like 3, 6, 9, 12. In all those cases where you have numbers increasing by 1 or 2 or 3, what that means is that your formula contains the expression n or 2n or 3n or something like that. Right? So on and so forth. So our general formula over here is going to include something like n. Now is this it? Is this the formula? Are we done? Well, you could always just plug in numbers just to sort of check and make sure that you're getting the right numbers in your sequence. The first term in the sequence is 5. So if I plug in n, equals 1 into this expression, all I'm going to get here is 1. So what's going to happen here is that we're oftentimes going to have to adjust our formula by adding, subtracting, multiplying, and dividing numbers or constants to get the sequence that we need. What you'll notice is that all of these numbers increase by 1 just like the index does, but the problem is that the first number isn't 1, it's 5. So what we have to do is that the first number is sort of shifted up by 4, so what you're going to have to do is you're going to have to add 4 to this, this formula over here. Now if you go ahead and check the first expression by adding 4, you're going to see that the number is 5, and that's exactly what the first term is. If you plug in n equals 2, you'll get 2+4, which equals 6. That's exactly what the second term is, and then so on and so forth. So what you'll see here is that n by itself wasn't enough. You're going to have to adjust your formula by adding a constant here. So that's the general formula for this sequence. Alright? Now let's move on to the second scenario here. A lot of times, you're going to see sequences that alternate signs. We've already seen an example of this with negative 1, positive 1, negative 1, positive 1. In our example here, we have negative 5, positive 5, negative 5, positive 5, so on and so forth. So we have the same number, but it flips the sign. Whenever that happens, that means you're going to have negative one raised to some power of n. Because what happens is as n goes bigger, you're going to have this exponent that oscillates between even and odd numbers, giving you even, giving you positive and negative numbers. And, really, what it comes down to is you're going to look at the first number in your sequence. And if it's negative, then you have this expression over here. If it's positive, then you have this expression over here. So in our example, the first sign that we see in this sequence is negative. So that means that our general formula is going to contain negative one raised to the n power. So is this all? Well, let's just go ahead and just sort of test some numbers here. A one, this is going to be negative one raised to the one power. Now that's just negative one. How do I get 5 out of this? Do I add a number to this or subtract the number? Well, what happens is if you add a number like, let's say, 4, then you're going to add 4, but then on the next term, it's going to be negative one plus 4, and that's not going to give you 5. So we can't add a number here, and, instead, what we're going to have to do is we're going to have to multiply a number. In order for the first number to be 5, I'm going to have to multiply this whole thing and adjust it by multiplying by 5. Now what you'll see here is that if you multiply this expression by 5, the first term will be negative one times 5, which is negative 5. The second term will be negative one squared, which turns to a positive times 5, and that equals positive 5. And then the whole thing will repeat over and over again. Right? So what you'll see here is that we had to adjust this formula by multiplying by 5, and this is the formula for your sequence. Let's move on to the 3rd situation here, which is that sometimes your sequences may contain fractions. So, in this sequence, we have 12, 23, 34, 45, 56. Whenever this happens, it means that your general formula is also going to contain fractions. And usually, what it's going to happen here is that the tops and the bottoms, the numerators and denominators are going to have slightly different patterns. So let's take a look at this, this situation over here. If you look at the numerators, which you'll see is that each one of these numbers, kind of like the first example that we did, they all increased by 1. It goes 1, 2, 3, 4, 5. And that's actually exactly what the n the index of your sequence does. Right? It also starts at 1 and it goes 1, 2, 3, 4, 5. So in this situation, when we have, things, you know, that increase by 1, it means that you're going to have an n inside of that expression. So in the numerator, this is actually just going to be n. And if you look at this, we actually won't even need to adjust it because for n equals 1, 2, 3, 4, 5, you'll exactly get these numerators, 1, 2, 3, 4, 5. Let's take a look at the bottoms here. This goes 2, 3, 4, 5, and 6. So just like the top, each one of these numbers also increases by 1, so so we know we're going to have to have an n here in the denominator. However, the starting number, kinda like this 5, 6, 7, 8, 9, isn't 1. It starts at 2 and then goes 3, 4, 5, 6. So we're going to have to adjust this a little bit, and we're going to have to do
Table of contents
- 0. Review of Algebra4h 16m
- 1. Equations & Inequalities3h 18m
- 2. Graphs of Equations43m
- 3. Functions2h 17m
- 4. Polynomial Functions1h 44m
- 5. Rational Functions1h 23m
- 6. Exponential & Logarithmic Functions2h 28m
- 7. Systems of Equations & Matrices4h 6m
- 8. Conic Sections2h 23m
- 9. Sequences, Series, & Induction1h 19m
- 10. Combinatorics & Probability1h 45m
9. Sequences, Series, & Induction
Sequences
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