If you have an original number of 3, your term numbers i would look like this top row. Subtract 0.1 once, subtract 0.1 twice, subtract 0.1 three times.Another way to think of it is that every time you need a new term, you multiply by 2. If you're looking at the if n is equal to, this is n is equal to four, well you're gonna subtract 0.1 three times, and you see that. The key thing is your'e starting at 9.6 and you're subtracting 0.1 one fewer times than the You could draw a table, and evaluate these if you want to. When n is equal to two, two minus one, you subtract 0.1 one time. When n is equal to one this term here is going to be zero, so this whole thing is gonna be zero. You can verify that this is going to work. If we're talking about the nth term, we subtracted this Whatever term we're talking about we subtract that term minus one times. If we're talking about the first term we subtract zero times. We gonna subtract it as a function of n? Let's see. We're gonna subtract 0.1, but how many times are We can say look, it's gonna be 9.6, but we're gonna subtract, we're gonna subtract 0.1 a certain number of times depending on what term So let's just call it, I don't know, let's just call it f of n. Create a function thatĬonstructs or defines this arithmetic sequence explicitly. Let's see if we can pause the video now and define this. You subtract a tenth you're gonna get 9.4, exactly what we saw over here. When h is three, it's gonna be h of two, h of two minus 0.1, minus 0.1. It's just gonna be this minus 0.1, which is going to be 9.5. When n is equal to two, we're now in this case over here, it's gonna h of two minus one, so it's gonna be h of one minus 0.1. Make a little table here, and we could say this is n, this is h of n, and you see when n is equal to one, h of n is 9.6. The second term is gonnaīe the previous term minus 0.1, so it's gonna be 9.5. One way to think about it, this sequence, when n is equal to one it starts at 9.6, and then every term is the previous term minus 0.1. Here we have a, we have a sequence defined recursively, and I want to create a function that defines a sequence explicitly. Let's do another example, but let's go the other way around. Get to the base case, which is when n is equal to one, and you can build up back from that. It all works out nice and easy, because you keep looking at previous, previous, previous terms all the way until you We're saying hey if we're just picking an arbitrary term we just have to look at the previous term and then subtract, and One and a whole number, so this is gonna be definedįor all positive integers, and whole, and whole number, it's just going to be the previous term, so g of n minus one minus seven, minus seven. Let me just write it, If n is equal to one, if n is equal to one, what's g of n gonna be? It's gonna be negative 31, negative 31. The first term when n is equal to one, if n is equal to one, In some ways a recursiveįunction is easier, because you can say okay look. Can we define this sequence in terms of a recursive function? Why don't you have a go at that. The next one is gonna be negative 52, and you go on and on and on. Start at negative 31, and you keep subtracting negative seven, so negative 38, negative 45. This is all nice, but what I want you toĭo now is pause the video and see if you can define If we're dealing with the second term we subtract negative seven once. If we're dealing with the third term we subtract negative seven twice. We subtract negative seven one less times than the term we're dealing with. What do we see happening here? We're starting at negative 31, and then we keep subtracting, we keep subtracting negative seven. It's gonna be negative 31 minus 14, which is equal negative 45. When n is equal to three, it's gonna be negative 31 minus seven times three minus one, which is just two, so we're gonna subtract seven twice. This is just going to be one, so it's negative 31 minus seven, which is equal to negative 38. It's going to be negative 31 minus seven times two minus one, so two minus one. This is just going to be zero, so it's going to be negative 31. H of n is negative 31, minus seven times one minus one, which is going to be. So I have a function here, h of n, and let's say that it explicitly defines the terms of a sequence.
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