 This video will talk about arithmetic sequences if a fixed constant is added to each successive term then we have an arithmetic sequence. So here's the definition. Given a sequence of a's of 1, a's of 2, a's of 3, these are just the different terms all the way up to the nth term where k and n are natural, notice I said natural we talked about that in the last video, numbers and k has to be less than n then if there exists a common difference such that a's of k plus 1 minus a's of k is equal to d for all k so then if that's true we have an arithmetic sequence. So what in the world does this mean? This really just means take the second term and subtract the first one a's of k plus 1 minus a's of k so you're going to take negative 2 minus 1 and that's going to give us negative 3 and we're going to take negative 5 minus a negative 2 which is really plus 2 and get negative 3 and negative 8 minus a negative 5 which is really plus 5 so we have negative 3 or negative 11 minus or plus 8 would give us negative 3, negative 14 plus 11 is going to give us negative 3 so the common difference is negative 3 and this is arithmetic. So we want to write the first 5 terms in arithmetic sequence given this information where we know we have a's of 1 is equal to and I'm going to rewrite this because it's kind of hard to see it so I'm going to write it as 1 half and d is equal to 0.25 or 1 fourth. Well we know the first one is 1 half and then we have to add 1 fourth to that. This is really then 2 over 4 so we have 3 over 4 so 3 fourths and then now we have 3 fourths and we again have to add 1 fourth so that's nice. We have 4 over 4 or just 1 and we have to add 0.25 again or 1 fourth more so if we take 4 over 4 and add another 1 over 4 we're going to have 5 over 4 and if we did it in fractions that would be fine it'd get it in decimals so it would really be like 0.5 and then 0.75 and then 1 and this would be 1.25 so you notice in the sequence that we just did the common difference was 0.25 we added 0.25 over and over again and repeatedly adding the same numbers really just multiplying that common difference like this we start with ace of 1 and we want 0 common differences we need it to stay 1 so this would be 1 plus 0 is 1. Well we can start with that first term again but this time I would like to have 1 common difference and that would give me 2 times 1 plus 1 or 3 but this is the most important thing that we're looking at right here is what is happening right here. So we take our first term and we add to that now 2 common differences and if I have my fourth term I would take my first term and I would add three common differences to it so what does that mean in general why I take my first term whatever that is and I add to that my common difference and then we have to figure out how many of them we need. Well if you look at this for the ace of 1 term I needed 0 ace of 1 I needed 0 ace of 2 I needed 1 ace of 3 I needed 2 ace of 4 I needed 3 so do you see the pattern happening here? How did these two numbers compare? 2 compared to 1 3 compared to 2 4 compared to 3 ace of n would be I'll put it over here this would be ace of n and we're letting this be if this was 1 and this was 0 1 less 2 minus 1 would be 1 3 minus 1 would be 2 4 minus 1 would be 3 so we're going to say n that's that little number down there minus 1 we need one less common difference than the term number and that's what we're going to say is our general formula ace of n let's put it up here again is the first term plus our common difference whatever that is times n minus 1 so we want to find the general term well what do we need we need an ace of 1 and we need a common difference ace of 1 is always your first term so we know that 7 they gave us our first term so we're good and then D is remember we're going to take 4 minus 7 and that's equal to negative 3 and we double check you take 1 minus 4 and that's again negative 3 and if I look at it looks like 1 plus negative 3 we get me to negative 2 negative 2 plus negative 3 we get me to negative 5 so it looks like my common difference is negative 3 then we have here ace of n is going to be equal to our first term 7 plus our common difference of negative 3 times n minus 1 well in your book they're one and they're going to want you to take it just a little bit further so they're going to want you to say 7 minus 3 n distribute the negative 3 and then negative 3 times negative 1 to be plus 3 so they really want you to say that ace of n is equal to negative 3 n plus 10 there's your nth term so the second part of this problem says find the sixth term well that means we want ace of 6 to negative 3 and then we put our 6 in here because that's the n and then plus 10 then that would give us negative 18 plus 10 or ace of 6 is going to be equal to negative 8 you may notice that when you look at these general terms they look very linear and then difference is actually equivalent to the slope of a line now I want you to understand something the n is a natural number and in linear this would be y equal m x plus b and x is a real number ace of n is like my y and n is like my x but the this is just a term number and this n is not all reals it's just natural numbers so if I were to grow and graph this this would be the line would be y equal negative 3 x plus 10 ace of n would be this point and this point and this point and this point and this point so ace of n equal negative 3 n plus 10 are just points there's a difference linear the line sequences are just points okay so now they're gonna ask us to find the number of terms so we need to know what that n is well we know that ace of n is equal to 42 and remember the formula again ace of n is equal to ace of 1 plus your d times n minus 1 let's plug and chug what we know ace of 1 is 4 and then plus d but we know that to be 2 and we don't know what the rest of that is and minus 1 because we're gonna try to solve for n but we do know on this side what ace of n is equal to it's equal to 42 okay so we just plugged in what we know now we're gonna just solve the equation I am going to simplify over here 42 is equal to 4 plus 2n minus 2 so 42 will be equal to 4 minus 2 would be 2 plus 2n to track the 2 to get it to the other side so I'd have 40 is equal to 2n and if I divide by 2 I find out the n is equal to 20 which means 20 terms and this is find the nth term we know the n is 20 ace 24 which is ace of 1 plus 2 times 20 minus 1 and we should get 42 if we did this correctly because we already know what ace of n is so 4 plus 2 times 19 and that's gonna give us 38 plus 4 which is gonna be 42 so we did it right so what else can we do with these arithmetic sequences again ace of n is equal to ace of 1 plus d times n minus 1 keep that in mind find the common difference d and the value of ace of 1 given all of this ace of 5 is equal to negative 17 ace of 11 is equal to negative 2 oh wow we don't really know a whole lot here do we so let's see if we can alter what's going on here let's say that we wanted to start with ace of 11 okay and I have to plug in as much as I can and then find one variable well I want to find first find that common difference so I can say this is equal to ace of 5 plus if I have ace of 5 ace of 11 was bigger and how many more common differences did I need I needed six more common differences and that's really because we started with 5 plus 6 will equal 11 this was the fifth term I needed six more common differences to actually be at the 11th term now let's plug and check what we know ace of 11 is negative 2 ace of 5 is negative 17 plus 60 and we're trying to solve for D as 17 to this side and you're gonna have 15 is equal to 6 D and dividing by 6 we have 15 divided by 6 but we can reduce that which will make the next part easier so if I reduce that they're both divisible by 3 so D is 5 over 2 now what do we do I got all kinds of things we can do but we kind of thought about this a little bit before I could say that I know let's say ace of 5 because it'll be smaller I don't know what ace of 1 is and I do know what my common difference is that's 5 halves times I have ace of 1 and I need ace of 5 so how many common differences do I need I need four of them okay or 5 minus 1 would be 4 so let's plug in what we know ace of 5 was negative 17 I could have used negative ace of 11 here as well it really doesn't matter which one I use ace of 1 is what I'm looking for plus when I multiply here I'm gonna have the 4 and the 2 cancel each other to have just 5 times 2 or 10 and if I subtract the 10 then I know that ace of 1 is equal to negative 27