Video transcript
- [Voiceover] So, this table here where you're given a bunch of Ns, N equals one, two, three, four, and we get the corresponding G of N. And one way to think aboutit is that this function, G, defines a sequence where Nis the term of the sequence. So for example, we couldsay this is the same thing as the sequence wherethe first term is 168, second term is 84, third term is 42, and fourth term is 21,and we keep going on, and on, and on. Now, let's think about whattype of a sequence this is. If we think of it as starting at 168, and how do we go from 168 to 84? Well, one way, you couldsay we subtract at 84, but another way to think about it is you multiply it by one half. So, times one half. And then to go from 84 to 42, you multiply by one half again. Times one half. And to go from 42 to 21, youmultiply by one half again. So, this right over hereis a geometric series. We're starting at a termand every successive term is the previous termtimes, it's often called the common ratio, times one half. So, how can we write Gof 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 functiondefinition that describes what we've just seen here starting at 168, and then multiplyingby one half every time you add a new term. Well, one way to thinkabout it is we start at 168, and then we're gonna multiply by one half, we're gonna multiply by onehalf a certain number of times. So, we could view the exponentas the number of times we multiply by one half. And how many times are wegonna multiply by one half? The first term, we multiplyby one half zero times. The second term, we multiplyby one half one time. Third term, we multiplyby one half two times. Fourth term, we multiplyby one half three times. So, the figure, it seemslike whatever term we're on, we're multiplying by one half,that term minus one times. And you can see that this works. If N is equal to one, you're going to have one minus one, that's just gonna be zero. One half to the zero's just one. So, you're just gonna get a 168. If N is two, well, two minus one, you're gonna multiplyby one half one time, which you see right over here, N is three, you're gonna multiply by one half twice. Three minus two is, or,three minus one is two. You're gonna multiply by one half twice, and you see that right over there. So, this feels like a reallynice explicit definition for this geometric series. And you can think of it in other ways, you could write thisas 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 overtwo to the N minus one. Another way you could think about it is, well, let's use our exponentproperties a little bit, we could say G of N isequal to, let's see, one half to the N minusone, that's the same thing as one half, let me write this. It's equal to 168. Lemme do this in a different color. So, this part right overhere is the same thing as one half to the N. So, times one half tothe N, times one half to the negative one. One half to the negative one. Well, one half to the negative one is just two, is just two, so, this is times two. So, we could rewrite this whole thing as 168 times two is what? 336? 336, did I do that right? 160 times two would be 320, plus 16, two times eight, so yeah, 336. And then times one half to the N. Times one half to the N. So, these are equivalent statements. This one makes a littlebit more intuitive sense, it kinda jumps out at you,you're starting at 168 and you're multiplying by one half. Whatever term you are minus one times. But this is algebraicallyequivalent to this, to our original one. But, can we also defineG of N recursively? And I encourage you to pausethe video and try to do that. In a lot of ways, the recursive definition is a little bit more straightforward, so let's do that. G, well, I'll make therecursive function a different, well, I got, I'll stickwith G of N since it's on this table right over here. G of N is equal to, and so, let's see, if we're going to, when N equals one, if N is equal to one,we're starting at 168. 168, and if N is greater than one and a whole number, so, if N, so, we're, this is gonna be definedover all positive integers, and whole number, what are we gonna do? Well, we're gonna takeone half and multiply it times the previous term. So, it's gonna be one halftimes G of N minus one. And you can verify that this works. If N is equal to one, wejust go right over here, it's gonna be 168. G of two is gonna beone half times G of one, which is, of course, 168. so, 168 times one half is 84. G of three is gonna beone half times G of two, which it is, G of three isone half times G of two. So, this is how we would define, this is the explicitdefinition of this sequence, this is a recursive functionto define this sequence.
