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- Our question asks us, how many toothpicks will be needed to
- form the 50th figure in this sequence?
- So let's look at this sequence.
- So the first figure in our sequence-- they look like
- little houses-- how many toothpicks are here?
- We have one, two, three, four, five, six toothpicks.
- So our first object here, or our first figure, has six
- toothpicks.
- How many do we have in our second figure?
- Well we're going to have these six that we had
- in the first, right?
- This is the first figure, right here.
- Let me just trace it.
- That's the first figure, right there.
- And then how many new toothpicks
- are we going to have?
- We have one, two, three, four, five.
- So we have six plus another five.
- So we have the six in the original one, plus another
- five, which is equal to 11 toothpicks.
- Now what happens in this third object, or this third little
- toothpick house looking drawing thing?
- So we have our figure from the two houses.
- So we have those right there.
- This is the exact same thing that we had
- as our second object.
- So that's that right there.
- And how many more do we have?
- Well it's going to be the same thing.
- We've just added on this other extension to our house.
- You can view it that way.
- So we have one, two, three, four, five.
- So it looks like every time we go down this sequence, or we
- add a new term to the sequence, or a new object to
- the sequence, we're adding five toothpicks.
- So here, we're going to have 11 plus 5.
- There's 11 toothpicks in this part of it, and then we
- have another 5.
- 11 plus 5, which is 16.
- Same thing over here.
- We have 16 in this part of the drawing.
- That part of the drawing is going to be 16 toothpicks, if
- you draw all of that.
- And then we're going to have another five right here.
- So this is going to be 16 plus 5 or 21.
- So how can we figure out how many toothpicks we're going to
- have in the 50th figure?
- I mean, we could draw 50 of them, but it's going to take
- us forever.
- So all we have to do is realize the pattern.
- Every one, we're just going to add 5.
- Or, we could even come up with a formula for the nth figure.
- How many toothpicks in the nth figure?
- So here, in the first figure, we have-- you could view it
- this way-- in this first figure, we have 1 plus 5
- toothpicks.
- Right?
- We have this 1.
- You could imagine that was always there, and then you
- added 5 right there.
- So that's our first term in our sequence.
- Our second term in our sequence, what do we have?
- We have 6, which is 1 plus 5.
- So we have that 1 plus 5, plus another 5.
- Plus this 5 right here, plus another 5.
- And then our third term in our sequence, what do we have?
- We have all of this business that we had
- in the second term.
- So 1 plus 5 plus 5, and then we have another 5.
- Let me do it in green.
- We have another 5.
- And then, finally, in this fourth term, what do we have?
- We have-- so the fourth one-- we have everything that we had
- in the third figure, so 1 plus 5 plus 5 plus 5.
- And then we add another 5.
- We add a fourth 5.
- We add a fourth 5 right there.
- So why did I do it this way?
- Because I wrote this first one as 1 plus 5 instead of just a
- 6 because you could say, well look, I had a 1, and
- then I had one 5.
- Now I have a 1, and I have two 5's.
- Now I have a 1 and I have three 5's.
- Now I have a 1 and I have four 5's.
- So you might see the pattern.
- If I had the nth term, if I had the nth term right here, I
- will have a 1 plus n 5's, right?
- The first term has one 5.
- The second term has two 5's.
- Third term has three 5's.
- Fourth term has four 5's.
- So the nth term, if this is, you know, if this is 10, it's
- going to be 1 plus 10 5's.
- Or if it's n, if I'm kind of abstracting a little bit, I'm
- going to have 1 plus 10-- sorry-- 1
- plus 5 times n, right?
- And try it out.
- If n is equal 1, it's going to be 5 times 1.
- If n is equal to 2, it's going to be 5 times 2.
- If n is equal to 3, it's going to be 5 times 3.
- So that when the 50th term, if we're talking about the 50th
- term, how many toothpicks are we going to have?
- Well we're going to have 1 plus 5 times 50, right? n is
- 50 here, which is equal to what?
- 5 times 50 is 250.
- That's 250 and then we add 1, we're going to have 251
- toothpicks will be needed to form the 50th
- figure in this sequence.