 Let's take a look at this AP problem. If the 5th term of an AP is 11 and the common difference is 2, then find the 11th term. Now when you look at this problem, you see these keywords AP and common difference in this term and that term and usually there's a formula that comes to your mind. It's called the nth term formula. What I want to do in this video is to not use that formula. Is to instead give you a more visual, more intuitive way to think about these problems. As a bonus, we'll get a better feel of where that formula actually comes from. So let's get started. As a first step, I'm going to bring in the number line. Now let's read the question again and see if we can put stuff from here on to our number line. We know that the 5th term is 11, so that means this over here a5, that's 11. We know that the common difference is 2, this means that the jump from 5th to 6th, that's going to be 2 units. In fact, any jump from one term to the next term, that's going to be 2 units. And we want to find the 11th term, this means this here is a question mark. That's something that we don't know. And now that we've put everything on this number line, I want you to pause the video and think about it. How would you move from this 5th term to this 11th term? Alright, if you take one jump, you go from the 5th term to the 6th term. If you take one more, you reach the 7th term. So how many jumps will you need to go from 5th term to this 11th term? Let's count together. This is one jump and then 2 and then 3 and then 4 and then 5 and then 6. So we need 6 jumps. I'm actually going to write 6 jumps here. And if each jump is 2 units, all of these jumps together will be 6 times 2 which is 12 units. And this is really useful because that's what we're doing when we're moving from the 5th term to the 11th term. Let me actually write this term. That 11th term that we want to find, that's actually our 5th term plus these 6 jumps. So if we know what these jumps are, which is 12 units and we know what the 5th term is, which is 11, we kind of know what our 11th term is. So 11th term becomes 11 plus 12, that's 23. This is our 11th term. And that's how I want you to think about these questions. You draw a number line, you put stuff from the question and you count the number of jumps. Let's try another problem to get a better feel of this. Here's the second problem. We know that the 9th term is 25 and the common difference is 3. We need to find the 4th term. Now I want you to pause the video and try this question on your own using this new number line method. All right, let's get started. We're given that the 9th term is 25, so in 9 is clearly 25. One jump is 3 units. That's what the common difference is. So if I take one jump from 9th to 10th, that's going to be 3 more units. In fact, that's going to be the case for every jump. And we need to find the 4th term. So this is where our question mark is. So this is what we have. We have the 9th term and we want to reach the 4th term. How would our jumps look like in this case? Now because our 4th term is to the left of A9, in this case we are going to jump left and not right. Let's see how many jumps we have. So from A9 we have one jump to A8 and then 2 and then 3 and then 4 and then 5. So here we have 5 jumps. But also know that these jumps are backwards. These are in the opposite direction. So I'm going to write 5 jumps backwards here. And if each jump is 3 units, 5 jumps will be 15 units. And because we're going backwards, it's going to take us 15 units back. So I can say that my 4th term is going to be where I start from, which is the 9th term, minus 15 units, which I get from jumping 5 times backwards. So my 4th term becomes A9, which is 25, minus 15. And I can say that my 4th term is 25 minus 15, which is 10. This is going to be my answer. Now I know what you're thinking about. You're probably saying to yourself that this number line thing works if you have to take a few steps here or a few steps there, but what if there are a lot of jumps? What if you can't put these numbers on the number line? What would you do then? Would you still use this number line? Let's take an example. Have a look at this one. If 17th term of an AP is 47 and the common difference is 4, find the 42nd term. And now we're really stuck. How do we visualize what's going on here without drawing a really, really long number line with 40 terms on it? Now would be a good time to pause the video and think about this one for a while. Where do we start? Where do we want to go? Where are the jumps? So let's ask the same questions that we asked for the last two ones. Where do we start? We started the 17th term. So our 17th term, that's 47. This is where we start. And where do we want to go? We want to go to 42nd term. So we want to come here, A42. This is our unknown. This is what we need to find. And the common difference here is 4. So we know that one jump is actually adding four units. Now asking the same question that we asked before, how many jumps do we need to move from 17th term to 42nd term? What's that going to be? Well, that's going to be the difference between this number 42 and this number 17. To move from 17 to 42, I'll have to take 42 minus 17 jumps. And that's going to give me, that's going to give me how much? Three here, two there, five and then 20. So 25 jumps. I need to take 25 jumps to move from 17 to 42 or 17th term to 42nd term. That's 25 jumps here. And if each jump is four, this means I'm going to add 25 times four, which is a very nice number. That's actually 100. I need to add 100 to my 17th term to get to my 42nd term. And if that's the case, then my 42nd term is going to be 100 plus 47, and that's 147. This is it. And this is the pattern that we see in all such problems. We start with a starting point, and then we have to reach a finishing point. And all we need to do is find out how many jumps we need to take from the starting point to the finishing point. And now let's take this idea of number line that we've just learned back to our nth term formula and see if we can make sense of that formula using this number line. So here's the problem. If the first term of an AP is A1 and the common difference is D, then find the nth term. Let's start with plotting things down. We know that the first term is A1. That's our first term. We know that the common difference, that's D, the common difference is D, and we need to find the nth term. So let's ask the same question to this problem as well. How many jumps do we need to take to reach the nth term? Well, if n was nine, we would take eight jumps. If n was eight, we would take seven jumps. It's always one less than n, and that's exactly what the formula says. Our nth term becomes An. That actually becomes your first term. This is where you start from A1 plus your n minus one, n minus one, minus one, and I'm writing this in blue so that you know where this one comes from, times each jump, which is D. That's the common difference. So now do you see where this formula comes from? It actually comes from starting with this first term, A1. But to be really honest, there's nothing special about this formula or this first term. If we start with some other term, we get a new formula. Let me show you what I mean. Let me start with a new term. Let's say I start with the fifth term, A5, and I want to find the nth term. How many jumps will I have to take if I start from the fifth term? Well, in this case, if our n is eight, I'm going to take three jumps, one, two, and three. That's eight minus five. If my nth term is nine, I'll take four jumps. If it's 10, I'll take five. It's always five less than where I want to be. So this time, my new formula becomes An. That's going to be equal to my new starting point, which is A5, plus my n minus five, n minus five, because I start with five, times my common difference. And this formula is as valid as the one that you keep seeing in the textbooks. Now, the formula that you use is not as important as knowing where it comes from. I have a feeling that if you get a hang of this number line, you can make your own formulae. And frankly, I feel that this number line is much more powerful than any formula that you see in this chapter.