Algebra

Patterns in sequences 2 | Linear equations | Algebra I | Khan Academy

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The question we are asked is how much toothpicks will be needed, to form the 50th figure in this row? So let’s look at the sequence. So the first figure in the row – the figures look like houses-how many sticks are there here? Here, one, two, three, four, five, six sticks. The first object here, or our first figure, contains six sticks. How many are in the second figure? Well, we’re going to have those six that we had in the first, right? This is the first figure here. Let me outline it. This is the first figure there. And how many new sticks we will have? Here, one, two, three, four, five. So we have six plus five more. So we have the six in the original figure, plus new ones five, which makes 11 sticks. And what happens in this third object, or in this little painted like a house of sticks? We have a figurine from both houses. Here they are, we see them there. This is exactly the same thing we had as our second object. This is that object there. How many more such sites do we have? Well it will be the same. We just added to the other extension to our house. It can be looked at this way. We have one, two, three, four, five. And as if every time we go down the line, or we add a new member to it, which is also a new object, we add five sticks. So here, we’re going to have 11 plus 5. In this part there are 11 sticks, after that we have 5 more. 11 plus 5, which gives 16. It’s the same here. In this part of the picture we have 16 sticks. This part of the picture will consist of 16 sticks if depict everything. And then we’ll have five more here. Which will be 16 plus 5, or 21. And how can we find how many sticks will we have in the fiftieth figure? I mean, we could draw 50 of the houses, but that would take us an etirnity. Ie all we have to do is realize the model. So we will only add 5. Or we can even find a formula for the nth figure. How many sticks are there in the nth figure? And here, in the first figure, we have – it can be seen so-in this first figure, we have 1 plus 5 sticks. Right? We have this 1. We can imagine she’s always been there, and we have added 5 more to the number. And this is the first member in our series. What will be the second member of the series? We have 6, which is 1 plus 5. So we have this 1 plus 5, plus 5 more. Plus 5 here, plus 5 more. Next is the third member of the line, what’s left for him? We have everything we had in the second article. Ie 1 plus 5 plus 5, and another 5. Let me paint it green. We have 5 more. And then, finally, in this fourth article, what do we have? We have – the fourth member – we have everything we had in the third figure, i.e. 1, plus 5, plus 5, plus 5. And we add 5 more. We add a fourth five. We add a fourth five there. And why did I do it that way? Because I wrote this first element as 1 plus 5, instead of like 6, because you can say, well, look, I had 1, and then I had 5. Now I have 1 and the fives are two. Then I have 1 and three fives. And then 1 and four fives. So maybe you see the model. If I have the nth member, if I have the nth member here, I will have 1 plus n fives, right? The first member has one five. The second term has two fives. The third member has three fives. The fourth member has four. Then the nth term, if this is, say, 10, he will be equal to 1 plus 10 fives. Or if it’s n, if we’re talking a little abstract, I will have 1 plus 10- sorry-1 plus 5 times n, right? And we try it. If n is equal to 1, it will be 5 times 1. If n is equal to 2, it will be 5 times 2. If n is equal to 3, we will have 5 times 3. So if the fiftieth article, if we talk about the fiftieth member, how many sticks will we have? Well, we will have 1 plus 5 times 50, right? n here is 50, which is equal to what? 5 times 50 is 250. We have 250, and we add 1, we have 251 sticks that will be needed to form the fiftieth figure in this row.

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