Over all positive integers, and whole number, what are we gonna do? Well, we're gonna take 168, and if N is greater than one and a whole number, so, if N, so, we're, this is gonna be defined 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, With G of N since it's on this table right over here. Recursive function a different, well, I got, I'll stick In a lot of ways, the recursive definition is a little bit more straightįorward, so let's do that. G of N recursively? And I encourage you to pause But this is algebraicallyĮquivalent to this, to our original one. You're starting at 168 and you're multiplying by one half. This one makes a littleīit more intuitive sense, it kinda jumps out at you, 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. Well, one half to the negative one is just two, is just two, so, this is times two. The N, times one half to the negative one. Here is the same thing as one half to the N. One, that's the same thing as one half, let me write this. Properties a little bit, we could say G of N isĮqual to, let's see, one half to the N minus Another way you could think about it is, well, let's use our exponent And you can think of it in other ways, you could write thisĪs G of N is equal to, let's see, one way you could write it, as, you could write it as 168,Īnd I'm just algebraically manipulating it over Nice explicit definition for this geometric series. You're gonna multiply by one half twice, and you see that right over there. If N is two, well, two minus one, you're gonna multiplyīy one half one time, which you see right over here, N is three, you're gonna multiply by one half twice. If N is equal to one, you're going to have one minus one, that's just gonna be zero. Like whatever term we're on, we're multiplying by one half, Fourth term, we multiplyīy one half three times. Third term, we multiplyīy one half two times. The second term, we multiplyīy one half one time. Gonna multiply by one half? The first term, we multiplyīy one half zero times. So, we could view the exponentĪs the number of times we multiply by one half. Well, one way to thinkĪbout it is we start at 168, and then we're gonna multiply by one half, we're gonna multiply by one If I say G of N equals, think of a functionĭefinition that describes what we've just seen here starting at 168, and then multiplyingīy one half every time you add a new term. 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. Times, it's often called the common ratio, times one half. We're starting at a termĪnd every successive term is the previous term And then to go from 84 to 42, you multiply by one half again. Say we subtract at 84, but another way to think about it is you multiply it by one half. If we think of it as starting at 168, and how do we go from 168 to 84? Well, one way, you could The first term is 168, second term is 84, third term is 42, and fourth term is 21,Īnd we keep going on, and on, and on. Say this is the same thing as the sequence where It is that this function, G, defines a sequence where N īooks VIII and IX of Euclid's Elements analyzes geometric progressions (such as the powers of two, see the article for details) and give several of their properties.- 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. It is the only known record of a geometric progression from before the time of Babylonian mathematics. It has been suggested to be Sumerian, from the city of Shuruppak. The general form of a geometric sequence isĪ, a r, a r 2, a r 3, a r 4, … ,Ī clay tablet from the Early Dynastic Period in Mesopotamia, MS 3047, contains a geometric progression with base 3 and multiplier 1/2. is a geometric sequence with common ratio 1/2.Įxamples of a geometric sequence are powers r k of a fixed non-zero number r, such as 2 k and 3 k. is a geometric progression with common ratio 3. In mathematics, a geometric progression, also known as a geometric sequence, is a sequence of non-zero numbers where each term after the first is found by multiplying the previous one by a fixed, non-zero number called the common ratio. The first block is a unit block and the dashed line represents the infinite sum of the sequence, a number that it will forever approach but never touch: 2, 3/2, and 4/3 respectively. Mathematical sequence of numbers Diagram illustrating three basic geometric sequences of the pattern 1( r n−1) up to 6 iterations deep.
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