Using inductive reasoning (example 2)

We're asked how many toothpicks will be needed to form the 50th figure in this sequence? And they've drawn the first four figures in the sequence. This is the first, the second, the third. Now, let's see what's happening here. So in this first figure, we have this nice little house that they've made out of toothpicks, and it has one, two, three, four, five, six toothpicks. So the first figure in our sequence has six toothpicks. Now, what happens in the second figure here? Well, it looks just like our first figure. We have our one, two, three, four, five, six toothpicks, but then they add some toothpicks. It looks like they added one, two, three, four, five, toothpicks. So it's 6 plus 5 or 11 toothpicks. So let me write this down. It is the 6 toothpicks plus 5 more, which is equal to 11 toothpicks. And then when we go to the third figure, it looks just like the second figure, so it has the 11 toothpicks, so the 11 toothpicks are one, two, three, four, five, six, seven, eight, nine, ten, eleven, just like the second figure. And then they add some more. They add one, two, three, four, five. So it looks like every time, when you draw part of a another house, because it can share a wall with the previous house or share a toothpick with the previous set of houses, you add five toothpicks. So for the third term in our sequence, we had 11 and when we had our second term, or our two houses, and then we add five more. So to our 50th term, so 11 plus 5 is going to be 16. And then for the fourth term, it's going to be 16 plus 5, which is equal to 21. Now, there's a couple of ways to think about it. You can think about however many more you are than 1. So let's say you are in the nth term of the sequence. If you are in the nth term, however more you are than 1, so if you're the nth term, you're going to be n minus 1 more than 1. If that is confusing, we'll do it with real numbers, so it gets a little bit more tangible. So the nth term, you are n minus 1 more or greater than-- I'll just say more-- than 1. For example, if n is 2, you are 1 more than 1. If n is 3, you are 3 minus 1, which is 2 more than 1. So however much more you are than 1, you multiply that by 5. We're 2 more than 1, so we add 10 to the number of toothpicks we have here. We're 3 more than 1 here, so we add 15 to the number of toothpicks we have there. So you could say that for the nth term, the number of toothpicks is equal to the number you're more than 1, < n minus 1 times 5. That's how much you're going to add above and beyond the amount of toothpicks in just the first sequence, so times 5. n minus 1 times 5 plus the number of toothpicks that you would just have in the first sequence or just this one house. And we already counted that: plus 6. So that's one way to think about it. And if this looks complicated, you just say, well, look, if I put a 4 here, I'm 3 more than 1, so 4 minus 1 is 3, so that'll be 3 times 5, which is 15. And then you add the number of toothpicks in 1 and then you get 6. Now another way, and many of you all might find this easier to think about, is even in 1, you could imagine a term here-- let me do it this way. You can imagine a term here that they didn't draw. You can imagine a 0th term. Let me just draw it here. Imagine a 0th term, and the 0th term would just be kind of a left wall of the house, or in this case, the left toothpick of a house. And then the first one, you're adding 5 toothpicks to that. In the second one, you're adding 5 toothpicks to that. And when you think about it this way, it actually becomes a little simpler to think in terms of n. Here, you could say, well, the nth term-- let me do this in a different color. You could say that the nth term is going to have-- or maybe we should say number of toothpicks in nth figure is going to be equal to 1. So in the 0th figure, which I just made up, you have at least 1 toothpick, and then whatever term in the sequence you are, you multiply that times 5 and you add that to the number of toothpicks in the 0th figure, so it's 1 plus 5 times n, which is actually a simpler way to think about. The first figure is going to be that 1 in the 0th figure plus 5: 6 toothpicks. Even in the 0th figure, it works out. If you put a 0 here, 0 plus 1 is 1. If you put the fourth figure here, 5 times 4 is 20 plus 1 is 21. Now let's answer their question. We need to figure out the toothpicks in the 50th figure in the sequence. Well, we just put 50 for n here. So for the 50th figure, we could use either formula. We have 1 plus 5 times 50. 5 times 50 is 250 plus 1 is equal to 251. And now I said you could use either formula because they should reduce to each other, if we did our math right or if we deduced properly. So this one up here, let's just verify this is the same thing as this over here. If we multiply 5 times n minus 1, we get 5n minus 5, right? We just distributed the 5. And then we have that plus 6. Negative 5 plus 6 is plus 1. So it's 5n plus 1 or 1 plus 5n. Either way, your 50th figure is going to have 251.