we are told that F of n is equal to F of n minus1 + 6 so the value of this function for each term n is defined in terms of the value of the function for previous terms we're essentially adding six to the previous term for each whole number n where n is greater than one and F of one is equal to 8 whenever you define something recursively like this where you're defining it in terms of a previous term you have to set up an initial point that you can start with and we'll see in a second why that's important now what I want you to do is pause this video and based on this definition figure out what the value of the function is for n equal 1 2 3 and four and then we're going to graph that and we're going to discuss that graph all right now let's work through this together and so let me in this column let we have n and here I will have F of n so we'll start with n equals 1 that's pretty straightforward they tell us that F of one is equal to 8 that was pretty straightforward now let's go to when n equals 2 well F of two is equal to F of 2 - 1 so it's equal to F of 1 + 6 + 6 well we know that F of one we just figured it out is8 so it's equal to 8 + 6 which is equal to 14 let's keep going maybe in purple all right so now we want to figure out what f of3 is going to be equal to well same idea it's going to be equal to F of 3 minus one or F of 2 + 6 we keep adding six every time so F of two we just figured out is 14 this is strangely fun 14 + 6 that is equal to 20 and then last but not least maybe in light blue when n equals 4 well let's figure out F of four is going to be equal to F of three + 6 which is equal to 20 F of 3 is 20 + 6 which is equal to 26 so you might have noticed a pattern here we start with when on our first term the value of the function is eight and then what did we do we added six and then to get to the next term we added six again and then we added six again and so we should see that visually when we actually try to graph it so let's graph it here and actually instead of calling this the x axis let me call this the N axis and the Y AIS let's just call that Y is equal to F of N and so let's take that first point when n equals 1 the value of our function is 88 it gets you right about there then when n is two we get to 14 2 14 right about there when n is three we get to 20 so that is there and then last but not least when n is four we get to 26 26 gets us right about there so you might notice something very interesting here it looks like these dots are on a line now this isn't a line because we're only defining this for whole number ends but we can see it looks like a line and every time we move forward by one we are moving up by six we move forward by one we're moving up by six so if this W were a line if I were to try to connect a DOT with or connect these dots with a line that line would have a slope of six because we our change in N is one and then our change in y or change in the value of our function is going to be six every time so in general if someone shows you a sequence like this and this is really an arithmetic sequence where each term is a previous term plus or minus some fixed amount you're going to see something that looks linear if you saw a curve then that wouldn't or something like dots on a curve then that wouldn't be an arithmetic sequence that would be something else but if you see dots that seem to form or be points on a line that's a pretty good clue that you're dealing with an arithmetic sequence