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- We are asked, what is the value of the 100th term in
- this sequence?
- And the first term is 15, then 9, then 3, then negative 3.
- So let's write it like this, in a table.
- So if we have the term, just so we have things straight,
- and then we have the value.
- and then we have the value of the term.
- I'll do a nice little table here.
- So our first term we saw is 15.
- Our second term is 9.
- Our third term is 3.
- I'm just really copying this down, but I'm making sure we
- associate it with the right term.
- And then our fourth term is negative 3.
- And they want us to figure out what the 100th term of this
- sequence is going to be.
- So let's see what's happening here, if we can discern some
- type of pattern.
- So when we went from the first term to the
- second term, what happened?
- 15 to 9.
- Looks like we went down by 6.
- It's always good to think about just how much the
- numbers changed by.
- That's always the simplest type of pattern.
- So we went down by 6, we subtracted 6.
- Then to go from 9 to 3, well, we subtracted 6 again.
- We subtracted 6 again.
- And then to go from 3 to negative 3, well, we
- subtracted 6 again.
- We subtracted 6 again.
- So it looks like, every term, you subtract 6.
- So the second term is going to be 6 less than the first term.
- The third term is going to be 12 from the first term, or
- negative 6 subtracted twice.
- So in the third term, you subtract negative 6 twice.
- In the fourth term, you subtract
- negative 6 three times.
- So whatever term you're looking at, you subtract
- negative 6 one less than that many times.
- Let me write this down just so-- Notice when your first
- term, you have 15, and you don't subtract
- negative 6 at all.
- Or you could say you subtract negative 6 0 times.
- So you can say this is 15 minus negative 6 times-- or
- let me write it better this way --minus 0
- times negative 6.
- That's what that first term is right there.
- What's the second term?
- This is 15.
- We just subtracted negative 6 once, or you could say,
- minus 1 times 6.
- Or you could say plus 1 times negative 6.
- Either way, we're subtracting the 6 once.
- Now what's happening here?
- This is 15 minus 2 times negative 6-- or, sorry
- --minus 2 times 6.
- We're subtracting a 6 twice.
- What's the fourth term?
- This is 15 minus-- We're subtracting the 6 three times
- from the 15, so minus 3 times 6.
- So, if you see the pattern here, when we have our fourth
- term, we have the term minus 1 right there.
- The fourth term, we have a 3.
- The third term, we have a 2.
- The second term, we have a 1.
- So if we had the nth term, if we just had the nth term here,
- what's this going to be?
- It's going to be 15 minus-- You see it's going to be n
- minus 1 right here.
- Right?
- When n is 4, n minus 1 is 3.
- When n is 3, n minus 1 is 2.
- When n is 2, n minus 1 is 1.
- When n is 1, n minus 1 is 0.
- So we're going to have this term right here is n minus 1.
- So minus n minus 1 times 6.
- So if you want to figure out the 100th term of this
- sequence, I didn't even have to write it in this general
- term, you can just look at this pattern.
- It's going to be-- and I'll do it in pink --the 100th term in
- our sequence-- I'll continue our table down
- --is going to be what?
- It's going to be 15 minus 100 minus 1, which is 99, times 6.
- right?
- I just follow the pattern.
- 1, you had a 0 here.
- 2, you had a 1 here.
- 3, you had a 2 here.
- 100, you're going to have a 99 here.
- So let's just calculate what this is.
- What's 99 times 6?
- So 99 times 6-- Actually you can do this in your head.
- You could say that's going to be 6 less than 100 times 6,
- which is 600, and 6 less is 594.
- But if you didn't want to do it that way, you just do it
- the old-fashioned way.
- 6 times 9 is 54.
- Carry the 5.
- 9 times 6, or 6 times 9 is 54.
- 54 plus 5 is 594.
- So this right here is 594.
- And then to figure out what 15-- So we want to figure out
- what 15 minus 594 is.
- And this can sometimes be confusing, but the way I
- always process this in my head is, I say that this is the
- exact same thing as the negative of 594 minus 15.
- And if you don't believe me, distribute out
- this negative sign.
- Negative 1 times 594 is negative 594.
- Negative 1 times negative 15 is positive 15.
- So these two statements are equivalent.
- This is much easier for my brain to understand.
- So what's 594 minus 15?
- We can do this in our heads.
- 594 minus 14 would be 580, and then 580 minus 1
- more would be 579.
- So that right there is 579, and then we have this negative
- sign sitting out there.
- So the 100th term in our sequence will be negative 579.