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.

thank you

Do you have any kids? I bet they are VERY VERY VERY VERY smart because you are! And i learned ALOT! Thank you Sal

I am in grade 7 and i am having trouble making an algebraic expression from a graph

Plz help

From patterns

what am i supposed to understand here

Wow! This is terrible!

this is actually really easy – he just made it seem difficult. Normally ka videos are good and simple. This isn't

you suck

🙁

what if the number of blocks for each terms arent all the same for example… term 1= 4 blocks term2= 9 term3= 16…how to get equation?

wtf I'm even more confused

you dont make me learn anything and i have a teast next week

This is great. I finally can do my project

I cant find an actual video about number patterns and sequences…..I have quiz tomorrow ahhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhh……my book is too advanced (New syllabus Primary Mathematics Singapore Math Worktext by rex) I am in 6th grade and the ones in my book are like VERY MUCH, TOO MUCH compared to the videos I can see!!!!!! UHH NOOOOOOO…….thanks tho.

number of squares = ( x – 1 ) * 4 + 1 = 4x – 4 + 1 = 4x – 3

A arithmetic sequence

?????

i am questioning my life now

i don get this