Explicit formulas for arithmetic sequences

so we see here in this table that for given ends when n is 1 f of n is 12 when n is 2 f of n is 5 when n is 3 f of n is negative 2 when n is 4 f of n is negative 9 and so one way to think about it is this function f is defining a sequence where the first term of this sequence is 12 the second term of the sequence is 5 the third term of the sequence is negative 2 the fourth term of the sequence is negative 9 it goes on and on and on and you might notice that it's an arithmetic sequence we start with a 12 and then the next term what have we done we've subtracted 7 now to go from the second to the third term what do we do we subtract 7 so each term is 7 less than the term before it now with that out of the way see if you can define this function this function of n if you can define it explicitly so figure out a function definition so I want to figure out F of n is equal I want you to figure out what this needs to be so that if you input and here it gives you the appropriate f of n so let's think about it a little bit it's going to be we could think of it as we're starting it the first term is going to be 12 but then we are going to subtract we're going to subtract 7 and what we're going to subtract 7 how many times we're going to subtract 7 so for the first term we subtract 7 0 times and so we just get 12 for the second term we subtract 7 once for the third term we subtract 7 twice 1 2 times for the fourth term we subtract 7 3 times so it looks like whatever term we're on we're subtracting 7 n minus 1 the we're subtracting 7 whatever term went on that term minus 1 times so it's n minus 1 times and let's see if this actually works out so f of 1 is going to be 12 minus 7 times 1 minus 1 that's a 0 so that's all just going to be 12 f of 2 is going to be 12 minus 7 times 2 minus 1 so we're going to so it's going to be 12 minus 7 times 1 or we're just going to seven once which is exactly the case we start at 12 and we subtract seven once F of three you can keep testing is 12 minus and we should have to subtract seven twice and we see three minus one is two times we're going to subtract seven two times so this looks right on we've defined the function explicitly we've defined F explicitly for this sequence let's do another example here so in this case we have some function definitions already here so you have your sequence you could kind of view it in this table or you could view it as the first term is negative 100 next term is negative 50 next term is 0 next term is 50 and it's very clear that this is also an arithmetic sequence we're starting at negative 100 and then what are we doing here we're adding 50 and then we're adding 50 and then we are adding 50 so each term is 50 more than the term before it and what I want you to do is pause the video and think about which of these definitions of the function f are correct and it might be more than one all right so let's let's think about it so this definition right over here one way to think about it it's saying okay I'm going to start at negative 100 and I'm going to add 50 n minus 1 times does this make sense well for the first term we're going to if we start at negative 100 we don't want to add 50 at all we want to add 50 zero times and it works out because 1 minus 1 is going to be zero so it checks out for N equals 1 let's see for N equals 2 you start negative 100 I want to add 50 once so I want to add I want to add 50 once so this should be a 1 2 minus 1 yep it's a 1 we're adding 50 whatever this number is whatever n is we're adding 50 one less that number that number of times so for here we're adding 50 twice for when n is 4 we're adding 50 3 times and this one checks out when n is 4 we're adding 54 minus 1 3 times negative 100 plus 50 times 3 or adding 53 times adding 50 1 2 3 times well that gives you 50 so I like this one now let's see about this one over here negative 150 plus 50 n all right that's one way of saying so let's see if we n is equal to one it's going to be negative let me do actually let me draw a table for this one so if we have N and we have F of N and this is going to be for this character right over here so if n is 1 it's going to be negative 150 plus 50 which is negative which is negative 100 yeah that checks out when n is 2 we get negative 150 plus 50 times 250 times 2 which is going to be this is 100 and there's negative 150 this is going to be negative 50 when n is 3 and that checks out of course 1 is 3 you get negative 150 plus 50 times 3 which is equal to 0 this checks out this one over here is going to work and you might say well hey these formulas look different but you can algebraically manipulate them see that there that to see that they are the exact same thing if you were to take this first one it's negative 100 plus let's distribute this 50 plus 50 n plus 50 n minus 50 minus 50 well negative 100 minus 50 that's negative 150 and then you have plus 50 n so these are algebraically the exact same definition for our function now what about this one here negative 100 plus 50 n does this one work let's see when n is equal to 1 this would be negative 100 plus 50 which is negative 50 well no this doesn't work we need to get a negative 100 here so this one is not not correct