Constructing arithmetic sequences
- [Instructor] We are asked what is the value of the 100th term in this sequence, and the first term is 15, then nine, then three, then negative three. 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 nine. Our third term is three. I'm just really copying this down, but I'm making sure we associate it with the right term. And then our fourth term, our fourth term is negative three. And they wanna ask, 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 we went from the first term to the second term, what happened? 15 to nine, looks like we went down by six. 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 six, we subtracted six. Then to go from nine to three, well we subtracted six again. We subtracted six again. And then to go from three to negative three, well we, we subtracted six again. We subtracted six again. So it looks like every term, you subtract six. So the second term is going to be six less than the first term. The third term is going to be 12 minus from the first term, or six subtracted twice. So in the third term, you subtract a six twice. In the fourth term, you subtract six three times. So whatever term you're looking at, you subtract six 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 six at all, or you could say you subtract six zero times. So you could say this is 15 minus six times or let me write it better this way, minus zero times six. That's what that first term is right there. What's the second term? This is 15, it's just we just subtracted six once, or you could say minus one times six. Or you could say plus one times negative six, either way, we're subtracting the six once. Now what's happening here? This is 15, this is 15 minus two times negative six, or sorry minus two times six, minus two times six. We're subtracting a six twice. What's the fourth term? This is 15 minus, we're subtracting the six three times from the 15, so minus three times six. So if you see the pattern here, when our term, when we have our fourth term, we have the term minus one right there, the fourth term we have a three, the third term we have a two, the second term we have a one. 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 one right here, right when n is four, n minus one is three. When n is three, n minus one is two. When n is two, n minus one is one. When n is one, n minus one is zero. So we're going to have, this term right here is n minus one, so minus n minus one times six. So if you wanna figure out the 100th term of this sequence, I didn't even have to write it in this general term, you could 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 gonna be what? It's going to be 15 minus 100 minus one, which is 99, times six, right? I just followed the pattern. One, you had a zero here, two, you had a one here, three, you had a two here, 100, you're gonna have a 99 here. So let's just calculate what this is. What's 99 times six? So 99 times six, actually you could do this in your head. You could say that's going to be six less than 100 times six, which is 600, and six less is 594. But if you didn't wanna do it that way, you just do it the old-fashioned way. Six times nine is 54, carry the five. Nine times six or six times nine is 54. 54 plus five is 594. So this right here is 594, and then to figure out what 15, so we wanna figure out, we wanna 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 one times 594 is negative 594, negative one 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? You should do it, we could do this in our head. 594 minus 14 would be 580, and then 580 minus one more would be 579. So that right there is 579, and then we have this negative sign sitting out there. So our the 100th term in our sequence will be negative 579.