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 sequences the first the second the third now let's see what's happening here so in this first figure we have this nice little little house that they made out of toothpicks and it has one two three four five six toothpicks so the first 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 looks like they added one two three four five toothpicks so it's 6 plus 5 or 11 - thanks so let me write this down it is the six toothpicks plus five more plus five more which is equal to 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 to the 11 to the 1 2 3 4 5 6 7 8 9 10 11 just like the second figure and then they add some more they add some more they add 1 2 3 4 5 so it looks like every time when you draw a part of another house because it can kind of 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 and I guess our two houses and then we add five more we add five more so to our 50th term so 11 plus five is going to be 11 plus five is going to be 16 and then for the fourth term it's going to be 16 plus five so it's going to be 16 plus five 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 one 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 one so if you're the nth term you're going to be n minus one more than one if that is confusing well we'll do it with real numbers so it gets a little bit more tangible so the nth term you are n minus one more or greater then I'll just say more then once for example if n is two you are one more than one if n is three 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 number of toothpicks number of toothpicks is equal to the number you're more than one so n minus 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 with 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 you get 6 now another way and many of y'all might find this easier to think about is even in 1 you could imagine you could imagine a term here well let me do it this way you could imagine a term here that they didn't draw you can imagine a 0th term so 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 and we're in this case the left toothpick of a house and then the first one you're adding 5 to thick stew that and the second when you're adding 5 to 2 that and we think about it this way it actually becomes a little simpler to think in terms of and here you could say well the nth term let me do this in a different color you can say that the nth term the nth term is going to have is going to have ordinary B we should say number of toothpicks an nth an nth figure number of toothpicks toothpicks in n figure is going to be equal to one so in the zeroth figure which I just made up you have at least one toothpick and then however whatever term in the sequence you are you multiply that times five and you add that to the number of toothpicks into the zeroth figure so it's one plus five times n which is actually a simple way to think about it the first figure is going to be that one in the zeroth figure plus five six toothpicks even the zeroth figure it works out if you put a zero here zero plus one is one if you put the fourth figure here five times four 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 50th figure we could use either formula we have one plus five times fifty five times fifty five 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 that's the same thing as this over here if we've if we if we multiply 5 times n minus 1 we get 5 n minus 5 right we just distribute to the 5 and then we have that plus 6 plus 6 negative 5 plus 6 is plus 1 plus 1 so it's 5 n plus 1 or 1 plus 5 n either way your 200 fit your 50th figure is going to have 251 toothpicks in it