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.