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Recursive and explicit sequences Video transcript - [Voiceover] So we've got the function g here, and what I want you to do is pause the video and figure out what g of one is, figure out what g of two is, g of three, and g of four. Figure out what all four of these are.
All right, now let's work through this together. So g of one, if n is equal to one, well, then we're gonna hit this case right over here, if n is equal to one, g is equal to four. So that was pretty straightforward. Now, g of two, if n is equal to two, well, two is greater than one and it's a whole number, and so we would use this case.
And this is interesting, because it's defined in terms of the function, but it's not defined in terms of g of n, but g of n minus one. So if n is two, 'cause we're evaluating g of two here, this is going to be g of two minus one, or g of one, plus 3.
Well, what's that going to be? Well, g of one, we know is equal to four. We just figured that out. So four plus 3. All right, let's keep going. G of three, we're gonna fall into this case again, because three is greater than one and it's a whole number.
So this is going to be g of three minus one, or g of two, plus 3. Well, we know what g of two is. And then g of four, well, we fall here again. This is gonna be g of three plus 3. G of three plus 3. What is that going to be equal to? Well, g of three we just figured out is And so what you have here, this is actually quite interesting.
You can think of this function g, and we see that it's defined over all positive integers. Because it's defined over positive integers, we could think of it as defining a sequence, and we see what the sequence here is.
The first term is four, second term is 7.
What's happening in this sequence? Well, we're starting with four. We're starting with four, and this case of the function gave us that. If n is equal to four, if n is equal to one, the function is going to be equal to four, and then for each term after that, you take the previous term and you add 3.Write arithmetic and geometric sequences both recursively and with an explicit formula, use them to model situations, and translate between the two forms.
★ While students work with both arithmetic and geometric sequences throughout high school, this is the sole standard in which students are explicitly asked to be able to translate between.
Write expressions in equivalent forms to solve problems. a. Determine an explicit expression, a recursive process, or steps for calculation from a context. b.
Combine standard function types using arithmetic operations. tracing paper, or geometry software. Specify a sequence of transformations that will carry a given figure onto another. Jan 27, · Here's an example of how to write an explicit and recursive equation for an arithmetic sequence. Scribd es red social de lectura y publicación más importante del mundo.
Functions are a very elusive concept for many students. They help us understand the world around us and are essential in the business world. This sort of sequence, where you get the next term by doing something to the previous term, is called a "recursive" sequence. In the last case above, we were able to come up with a regular formula (a "closed form expression") for the sequence; this is often not possible (or at least not reasonable) for recursive sequences, which is why you need .