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# Intro to arithmetic sequences

CCSS.Math:

## Video transcript

what I want to do in this video is familiarize ourselves with a very common class of sequences and this is arithmetic arithmetic sequences and they're usually pretty easy to spot their sequences where each term is a fixed number larger than the term before it so my goal here is to figure out which of these sequences or arithmetic sequences and then just so that we have some practice with some of the sequence notation I want to define them in either Express explicit functions of the term you're looking for the index you're looking at or as recursive definitions so first given that an arithmetic sequence is one where each successive term is a fixed amount larger than the previous one which of these are arithmetic sequences well let's look at this first one right over here to go from negative 5 to negative 3 we had to add two then to go from negative 3 to negative 1 you have to add two then to go from negative 1 to 1 you had to add 2 so this is clearly an arithmetic sequence we're adding the same amount every time and there's several ways that we could define the sequence we could say it's a sub N and you don't always have to use K this time I'll use n to denote our index from N equals 1 to infinity with and there's two ways we could define it we could either define it explicitly or we could define it recursively so if we wanted to define it explicitly we could write a sub n is equal to whatever the first term is in this case our first term is negative 5 it's equal to negative 5 plus we're going to add to 1 less times than the term we're at so for the second term we add 2 once for the third term we add 2 twice for the third for the fourth term we added from our base term we added 2 3 times so we're going to add 2 we're going to add positive 2 1 less than the index that we're looking at n minus 1 times so this is an explicit definition of this arithmetic sequence if I wanted to write it recursively I could say a sub 1 is equal to negative 5 and then each successive term for a sub to and greater so I could say a sub n is equal to a sub n minus 1 plus 3 each term is equal to the previous term plus or not 3 plus 2 so this is 4 for N greater than or equal to 2 so either of these either of these are completely legitimate ways of defining of defining the arithmetic sequence that we have here we can either define it explicitly or we could define it recursively now let's look at this sequence is this one arithmetic well we're going from 100 we add 7 107 114 or adding 714 to 121 we are adding 7 so this is indeed an arithmetic sequence just to be clear this is one and this is one right over here and we could write that this is the sequence a sub N and going from 1 to infinity of and we could just say a sub n if we want to define it explicitly is equal to 100 plus we're adding 7 every time and then each term the second term we added 7 once third term we add 7 twice so for the nth term we're going to add 7 n minus 1 times so this is an explicit definition of it and what we could also do it recursively we could also say we could also so just to be clear this this is one definition where we write it like this we write it like this or we could write a sub n from N equals 1 to infinity and either case I should write with with and if I want to define it recursively I could say a sub 1 is equal to 100 and then for a sub n for anything larger than 1 for any index above 1 a sub n is equal to the previous term plus plus 7 and once so we're done this is another way of defining it so in general if you wanted a generalizable way to a spot or defined an arithmetic sequence you could say an arithmetic sequence is going to be of the form a sub n if we're talking about an infinite 1 from N equals 1 to infinity if you wanted to find it explicitly you could say a sub n is equal to is equal to some constant which would essentially be the first term it would be some constant plus plus some number that you're incrementing or I guess this could be a negative number or decrementing by times n minus one so this is one way to define an arithmetic sequence in this case D was two in this case D is seven that's how much you're adding by each time and in this case K is negative five and in this case K is 100 the other way that you could if you wanted to write the recursive way of defining an arithmetic sequence generally you could say a sub one is equal to K and then a sub n is equal to a sub n minus one a given term is equal to the previous term plus D for N greater than or equal to two so once again this is explicit this is a recursive way of defining it and we would just write with there now the last question I have is is this one right over here an arithmetic sequence well let's check it out to start we start one then we add two then we add three so this is immediate give away this is not an arithmetic sequence now we are adding four we're adding a different amount every time so how could we so this first of all not this is not arithmetic this is not an arithmetic sequence but how could we define this since when we're trying to define our sequences so if we wanted to define it if we wanted to define it let's say listen we wanted to define it recursively so this could we could say this is equal to a sub n where n is starting at one and it's going to infinity with with will we'll say our base case a sub one is equal to one and then for n is two or greater a sub n is going to be equal to what so a sub two is the previous term plus two a sub 3 is a previous term plus 3 a sub four is the previous term plus four so it's going to be the previous term it's going to be the previous term plus whatever your index is Plus whatever your index is so this looks close but notice here we're changing the amount that we're adding based on what our index is we're adding the amount of index to the previous tournament so this is four and is greater than or equal to two well for an arithmetic sequence we're adding the same amount regardless regardless of what our index is here we're adding the index itself so this one is not arithmetic but it's an interesting sequence nonetheless