Equations of Sequence Patterns Equations of Sequence Patterns
Equations of Sequence Patterns
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- 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.
- 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
- 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.
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