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Current time:0:00Total duration:8:36

Sequences word problem: growth pattern

Video transcript

our question asks us what equation describes the growth pattern of this sequence of blocks 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 one block right there so let me write the term right up here so I have the term term and then I'll have the number of blocks number of blocks 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 for block so it's like one plus four right there so we're going to have five blocks right there we added four 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're 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 it 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 gonna 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 one let's say that X is equal to the term we add just this one there then when X is equal to two we added one column of four so this is when X is equal to two we have one column of four then when X is equal to three we have two columns of four right there and you could even say when X is equal to one you had zero columns right we had no nothing no extra disk columns of four blocks we didn't have any and then when X is equal to four we had three columns we had three columns there when X is equal to four so what's a pattern here or Wow 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 box so let me write it this way if I write the number of blocks let me write it this number number of the block of blocks 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 it 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 one we have no columns of four blocks when X is equal to two we have one column when X is equal to three 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 it's going to be X minus one right when X is two 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 1 2 3 columns here we have 1 2 columns here we only have one column here we have 0 column 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 the number of columns and then in each column we have 4 blocks so it's the number of columns x times 4 right for each of these columns we have 1 column we have 1 2 3 4 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 the number of blocks and then we could simplify this we have a 1 and we have a minus 4 I guess you were subtracting 4 from it so this is going to be equal to 4 X minus 3 is the number of blocks given our X terms if we four 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 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 4 blocks so we could write it this way we could write change so this triangle right here means change Delta means change in blocks change in blocks change in blocks divided by divided by change in X now you might recognize this this is slope so and if you don't worry you know if you don't if slope is a completely foreign concept to you you can just do it the way we did the first part of this video and that's a completely legitimate way and hopefully it'll 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 at 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 4 is change in X is 4 our slope is equal to 4 so if you want to do this kind of the setting up align the equation of a line way you would say that our equation if if well let me write it number of blocks number of blocks are going to be equal to four times our four times the the term that we're dealing with the you know the 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 and our first term we only had one block so let's put that here so in our first term so when our first term we're going to have that B right there we only had one block so we have one is equal to four plus B if you subtract four from both sides of this equation so you subtract four from both sides what do you get on the left-hand side 1 minus 4 is negative 3 and that's equal to these fours cancel out and that's equal to B so another way to get the equation of line we've just solved that B is equal to negative 3 we said how much do the chip blocks the number of blocks change for a certain change in X it's just a change in the number of blocks for a change in X we saw it's always 4/4 / 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 non I guess positive integers in this situation you could view this as the 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 is back here you get 4 X minus 3 which is what we got earlier in the video right there hopefully you found that fun