do you remember when you were given a sequence and you were trying to figure out the formula to get that sequence but it just wouldn't come to you I'm going to make your life so much easier by giving you a template to get that equation okay but this particular template only deals with arithmetic sequences so I think I should probably tell you what an arithmetic sequence is first an arithmetic sequence is a type of sequence where the difference between the consecutive numbers is a constant what does that mean in English Okay so here is an arithmetic sequence and uh notice that it is going up by sixes each time we're adding an arithmetic sequence is where you're going to add the previous term by a constant number every single time to get your next consecutive term okay so what I'm going to do is I'm actually going to use this example to explain the template this TN represents your term value let's say for instance this 11 is the second term so when my n or my term number is two I'm going to get 11 as my term value okay now your a is your very first term in the sequence and your D was that six that was going up each time I also want you to note that if your sequence did go down you could be adding a negative number in which case your D would be a negative okay so it is possible to go down as well now let's just see if you do get the 11 so two your second term minus 1 gives you 1 so this 1 * your six gives you six so this whole guy equals to six plus your first term which was five does give you the 11 okay so it does work now the rest of the tutorial is going to be going over very common types of questions that you'll see and the very first one is here's a sequence give me the formula to describe the nth term so that if I asked you for say the 12th term you could use that formula and figure out what the 12th term's value is okay so I put the answer down but I'll go through it with you they gave me a little bit of an arithmetic sequence and they've actually told me that it was an arithmetic sequence so unless they um didn't give me any indication that was an arithmetic sequence I might not be able to use this formula I'd have to use a different template okay so I know my a value is nine because that's the first number in the sequence and it also looks like it's going up by fours each time and that's why I put D equals to 4 then I just plug them into the template so I'm looking for TN in I guess you could think of it as your dependent variable your n's going to be your independent variable and here's my nine and my four remember to put brackets around the four okay then you're going to do distributive property and you get 4 n minus 4 and then that 9 in the front then you're going to simplify so you get your 4N and then the 9 minus the four gives you the positive five so this is my formula Al together it's almost like my X and my y so that if they ever told me hey find out what Y is when X is 12 then you're going to put 12 into the formula and you're going to figure out what your Y is so the 12th term is going to give you a value of 53 okay and there you go here's the equation and the 12th term what if we were given say a recursion formula they told us the first term five they even told us the second term 11 so it kind of looks like it's going up by sixes but they didn't tell us any other terms so we can't really say it is an arithmetic sequence and if it's not again we can't use this formula so what we have to do is we kind of have to maybe find one or two more terms just to make sure it is arithmetic okay so what I did was I took this recursion formula and I decided to find T3 so I put T3 minus 1 right here + 6 and that gives me T2 + 6 and I know what T2 is it's 11 they gave that to me already okay and I get 17 so here's my first term my second term and then my third term so yes it does look like it's going up by sixes and we can say that it's arithmetic since it's arithmetic we can then use the formula so let's create an overall explicit formula that describes this um sequence okay so here's the first term your D is going up by sixes so positive six and then again we're just going to plug those into the formula expand and simplify and we get TN = to 6 nus1 if we wanted to find the 12th term again we're just going to plug 12 into our n and then we're going to get our T12 value which is 71 it's really easy to actually just plug this into our c um the calculator and just press equals a bunch of times after you add six and see if you do land on the 12th number as 71 okay so here's the next type of question given a sequence that ends so they actually gave you part of a sequence and then the ending number you might have to determine how many numbers are in that sequence in other words how many terms are there okay so we know that this is the first term and that's the second and that's the third but what is this like what term number is that I don't know this is just a term value okay so what we're going to do is we're going to figure out what we know I know my a it's right there -3 it seems to be going up by the same amount which is five okay and we can always do that by taking one of them subtracting the one before it and getting five I don't know what my n is but I do know that this last term is probably my nth term so this is TN my nth terms value here's my arithmetic equation and I'm going to plug in what I know so this is 152 that was -3 and this is five then all you have to do is you solve for n so do distributive law and then simplify bring the eight over to the other side divide by five and it's all simple arithmetic okay so you're always looking for what was given to you what are you trying to find the last type of question is a little bit confusing to people but it's not too bad once you get the hang of it see if you are given two random terms you want to figure out what TN is so what is a formula to describe this sequence they have to tell you that it's an arithmetic sequence because other than that you don't have enough information so if you know that it's an arithmetic sequence you know that you can use these two formulas okay let's just deal with this one first they've told you that the sth term is 36 so that means that if my n is 7 my TN is going to be 36 I don't know what my a is cu they never told us and I also don't know what my D is um because they didn't tell us that either so unfortunately I have my a and my D still there this is going to be called my first equation then I'm going to make another equation with this guy and I'm going to do the exact same thing so the 15th term or when n is 15 I'm going to get a term value of 68 okay and I'm just going to simplify a little bit get 14 and again I still have A's and D's this is going to be my second equation now what I'm going to do is I'm just going to use grade 10 I guess math knowledge and I'm just going to do substitution or elimination I figure since uh the A's are the same there's no numbers in front of the A's let's just do Elimination we'll just line them up so we have a + 6D like this and I decide to put the 36 on this side and then I got a + 14 D and that's right there decid to do the 68 on this side and I subtract it so then a minus a is going to be gone I have no A's 60 minus 14d I get 8D equal 36 - 68 and I get - 32d then I divide both sides by8 to get rid of that and D equal to 4 that means that I know that each of my terms goes up by fours then what I'm going to do is I'm going to start solving for my a so I'm just going to plug get back into whichever equation I want and I'm thinking maybe this one looks a little bit easier it has smaller numbers so I have a + 6D a + 6 and I know my D is 4 equals to 36 so that's where that 36 comes from and 6 fours is 24 bring it over and I get a is 12 so my very first term is 12 my difference is four and I can now create an equation so I'm going to put 12 in for my a because I have it right there and we're not going to put anything in there because we just want a random um explicit formula and then we have our D which is four and again we're just going to expand and simplify there's my explicit formula to describe this particular um sequence