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Sequences word problem: growth pattern

Sal finds the equation that describes a growth pattern of shapes made of squares. Created by Sal Khan and Monterey Institute for Technology and Education.

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Video transcript

Our question asks us, what equation describes the growth pattern of this sequence of a block? So we want to figure out, if I know that x is equal to 10, how many blocks am I going to have? So let's just look at this pattern here. So our first term in our sequence, or our first object, or our first pattern of blocks right here, we just have 1 block right there. So let me write, the term-- write it up here --so I have the term and, then I'll have the number of blocks. So in our first term, we had one block. And then our second term-- I'll just write this down, just so we have it --what happened here? So it looks just like our first term, but we added a column here of four blocks. So it's like 1 plus 4 right there. So we're going to have five blocks right there. We added 4 to it. Then in our third term what happened? What happened in our third term? Well it just looks just like the second term, but we added another column of four blocks here. Right? We added this column right there. If you imagine they were being added to the left-hand side of the pattern. So we added four more blocks. We have nine blocks now. We have nine blocks, so it looks like each time we're adding four blocks. And on this fourth term, same thing. The third term is just this right here. This right here is what the third term looked like, and then we added another column of four blocks right here. So we added four more, so we're going to have 13 blocks. So our fourth term is 13. So let's see if we can come up with a formula, either looking at the graphics, or maybe looking at the numbers themselves. So one way to think about it, so we start off with-- So when x is equal to 1, let's say that x is equal to the term, we add just this 1 there. Then when x is equal to 2, we added one column of four. So this is when x is equal to 2, we have one column of four. Then when x is equal to 3, we have two columns of 4, right there. And you could even say when x is equal to 1, you had zero columns, right? We had no, nothing, no extra columns of four blocks. We didn't have any. And then when x is equal to 4, we had three columns. We had three columns there, when x is equal to 4. So what's the pattern here? Or how can we express the number of blocks we're going to have, given the term that we have? Well, it looks like we're always going to have one block, so let me write it this way. If I write the number of blocks-- let me write it this way --it looks like we're always going to have one, right? We have this one right here, that one right there, that one right there, that one right there. Looks like we always have one plus a certain number of columns of four, but how many columns do we have? When x is equal to 1, we have no columns of four blocks. When x is equal to 2, we have one column. When x is equal to 3, we have two columns. So when x is equal to anything, it looks like we have one less number of columns. So it's going to be x minus 1, right? When x is 2, x minus 1 is 1. When x is 3, x minus 1, so this right here is x minus 1. x is 2, this is x minus 1. This is x minus 1. This is x minus 1, and x minus 1 will tell us the number of columns we have, right? Here we have one, two, three columns. Here we have one, two columns. Here we only have one column. Here we have zero columns. So it even works for the first term. And in every one of these columns, so this right here, x minus 1 is the number of columns, and then in each column we have four blocks. So it's the number of columns times 4, right? For each of these columns, we have one column. We have one, two, three, four blocks. So this is the equation that describes the growth pattern. So let me write this, let me simplify this a little bit. If I were to multiply 4 times x minus 1, I get the number of blocks being equal to 1 plus 4 times x. I have to distribute it. 4 times x is 4x, and then 4 times negative 1 is negative 4. So that's equal to the number of blocks. And we could simplify this. We have a 1 and we have a minus 4, or I guess we're subtracting 4 from it, so this is going to be equal to 4x minus 3 is the number of blocks given our x term. So if we're on term 50, it's going to be 4 times 50, which is 200 minus 3, which is 197 blocks. Now another way you could have done it is you could have just said, look, every time we're adding 4, this is a linear relationship, and you could essentially find the slope of the line that connects this, but assume that our line is only defined on integers. And that might be a little bit more complicated, but the way that you think about it is, every one, every time we added a block, we added-- or every time we added a term we added four blocks. So we could write it this way. We could just write change-- so this the triangle right here means change. Delta means change in blocks divided by change in x. Now you might recognize this. This is slope. And if you don't worry, if slope is a completely foreign concept to you, you can just do it the way we did it the first part of this video. And that's a completely legitimate way, and hopefully it will make some connections between what slope is. So what is the change in blocks for a change in x. So when we went from x going from 1 to 2-- so our change in x here would be 2 minus 1, we increased by 1 --what was our change in blocks? It would be 4, or 5 minus 1. It's 5 minus 1. And what is this equal to? This is equal to 4 over 1, which is equal to 4. Let me scroll over a little bit. So our change in blocks, or change in x is 4, or our slope is equal to 4. So if you want to do this kind of the setting up the equation of a line way, you would say that our equation-- If, well let me write it. Number of blocks are going to be equal to 4 times the term that we're dealing with, the term in our pattern, plus some constant. This right here is the equation of a line. If it's completely foreign to you, just do it the way we did it earlier in the video. And so, how do we solve for this constant? Well, we use one of our terms here. We know that when we had one-- In our first term we only had one block. So let's put that here. So in our first term-- we're going to have that b right there --we only had one block. So we have 1 is equal to 4 plus b. If you subtract 4 from both sides of this equation, so you subtract 4 from both sides, what do you get? On the left-hand side, 1 minus 4 is negative 3, and that's equal to-- these 4's cancel out --and and that's equal to b. So another way to get the equation of a line, we have just solved that b is equal to negative 3. We said how much do the number of blocks change for a certain change in x, which is a change in the number blocks for a change in x, we saw it's always 4. 4 per change in x. When x changes by 1, we change by 4. That gave us our slope. And then to solve for-- If you view this as a line, although this is only defined on integers, I guess positive integers. In this situation, you could view this as a y-intercept. To solve for this constant, we just use one of our terms. You could have used any of them. We used 1 and 1. You could use 3 and 9. You could use anything. We solved b is equal to negative 3, and so if you put b back here, you get four x minus 3, which is what we got earlier in the video, right there. Hopefully you found that fun.