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# Arithmetic sequence problem

Video transcript
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. And then to go from 3 to negative 3, well, 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.