Explicit & recursive formulas for geometric sequences

so this table here we are given a bunch of ends n equals 1 2 3 4 we get the corresponding G of N and one way to think about is that this function G defines a sequence where n is the term of the sequence so for example we could say this is the same thing as the sequence where the first term is 168 second term is 84 third term is 42 and forth for fourth fourth term is 21 and we keep going on and on and on now let's think about what type of a sequence this is if we see little if we think of as starting at 168 and how do we go from 168 to 84 well one way you could say we subtracted 84 but another way to think about it is you multiplied by 1/2 so times 1/2 and then to go from 84 to 42 you multiply by 1/2 again times 1/2 and then go from 42 to 21 you multiply by 1/2 again so this right over here is a geometric series we're starting at a term and every successive term is is the previous term times it's often called the common ratio times 1/2 so how can we write G of n how can we define this explicitly in terms of N and I encourage you to pause the video and think about how to do that so construct a so if I say G of N equals think of a function definition that describes what we've just seen here starting at 168th and then multiplying by 1/2 every time you add a new term well one way to think about it is we start at 168th and then we're going to multiply by 1/2 we're going to multiply by 1/2 a certain number of times so we could view at the exponent as the number of times we multiply by 1/2 and how many times are we going to multiply by 1/2 the first term we multiply by 1/2 zero times the second term we multiply by 1/2 one time third term we multiply by 1/2 2 times fourth term we multiply by 1/2 three times so figure it seems like whatever term we're on we're multiplying by 1/2 that term minus one times and you can see that this works if n is equal to one you're going to have 1 minus 1 that's just going zero 1/2 to the 0 is just 1 so you're just going to get a 168 if n is 2 well 2 - what you're going to multiply by 1/2 one time which you see right over here n is 3 you're going to multiply by 1/2 twice 3 minus 2 is or 3 minus 1 is 2 you're going to multiply by 1/2 twice and you see that right over there so this feels like a really nice explicit definition for this geometric series and you can think of it in other ways you could write this as G of n is equal to let's see one way you could write it as you could write it as 168 and I'm just algebraically manipulating it over 2 to the N minus 1 another way you could think about it is well let's let's let's use our exponent properties a little bit we could say G of n is equal to let's see 1/2 to the N minus 1 that's the same thing as 1/2 let me write this it's equal to 168 let me just in a different color so this part right over here is the same thing as 1/2 to the N so times 1/2 to the N times 1/2 to the negative 1 1/2 to the negative 1 well 1/2 to the negative 1 is just 2 is just 2 so this is times 2 so we could rewrite this whole thing as 168 times 2 is what 336 336 I do that right 160 times 2 would be 320 plus 16 2 times 8 so yeah 336 and then times 1/2 to the N times 1/2 to the N so these are equivalent statements this one makes a little bit more intuitive sense it kind of jumps out at you you're starting at 168 and you're multiplying by 1/2 whatever term you are minus 1 times but this is algebraically equivalent to this to our original one now can we also define G of n recursively and I encourage you to pause the video and try to do that and in a lot of ways the recursive definition is a little bit more straightforward so let's do that G well I'll make the recursive function a different well I'll still stick with G of n since it's not on this table right over here G of n is equal to so let's see if we're going to when N equals 1 if n is equal to 1 we're starting at 168 168 and if n is greater than 1 and a whole number so if n so we're this is going to be defined over all positive integers and whole number what are we going to do well we're going to take 1/2 and multiply it times the previous term so it's going to be 1/2 times G of n minus 1 and you can verify that this works if n is equal to 1 we just go right over here it's going to be 168 G of 2 is going to be 1/2 times G of 1 which is of course 168 so 168 times 1/2 is 84 G of 3 is going to be 1/2 times G of 2 which it is G of 3 is 1/2 times G of 2 so this is how we would define this is the explicit definition of this of the sequence this is a recursive function to define this sequence