![]() ![]() Illustrated definition of Geometric Sequence: A sequence made by multiplying by the same value each time. (each number is 2 times the number before it) Sequence. Hits the point home that there's multiple ways, either with a traditional, I guess you would say explicit function or a recursive function like this.\( \newcommand\) which makes sense if you think of the triangular numbers as counting the number of handshakes that take place at a party with \(n+1\) people: the first person shakes \(n\) hands, the next shakes an additional \(n-1\) hands and so on. A sequence made by multiplying by the same value each time. And now instead of saying n is greater than or equal to zero, now n is greater than or equal to one. Write a different domain, where n still has to be an integer. Zero is equal to six, we could write t of one is equal to six. Have kept all of that, or I'm gonna have to rewrite all of that. You could write it, actually, maybe I should That it generates the six when n equals one, youĬould do it this way. In that case, it'd be t of, or, sorry, it would be two When you get n equals one, t of one is going to be two times t of one minus one, t of zero. And then this is going to be for, or maybe I'll write it this way, where n is an integer and n is greater than or equal to zero. And then we could say t of n is equal to two times t of n minus one, t of n minus one. So we could, if we want a recursiveĭefinition for the sequence, we can define the first term, or, in this case, weĬould say the zeroth term if we want to start at n equals zero. We could say, all right, look, it looks like each of these terms in our sequence is So notice, these areĭifferent function definitions with different domains, but they're generating Then when n is equal to two, it's six times two to the two minus one, which is just two to the first power. So what happens? One minus one, we get that zeroth power that we want right over there. N equals one in here, we could maybe call this the first term. So I could say six times two to the n minus one power, where n is an integer and n is greater than or equal to one. How do I do that? Well, I just subtract one from it. So what you could do is, is when you input a one, thisĮssentially becomes a zero. This as the zeroth term, I want to start at n equals one. For the exact same reason allowing the zero sequence as a geometric sequence is somewhat problematic (no unique ratio can be determined for it) but it is OK (in fact necessary) to include the zero sequence in the subspace of geometric sequences with given ratior. Gonna start at n equals zero, and you could kind of view ![]() Answer: The sum of the given arithmetic sequence is -6275. So we have to find the sum of the 50 terms of the given arithmetic series. There we found that a -3, d -5, and n 50. And let's say I want to, instead of saying, okay, I'm This sequence is the same as the one that is given in Example 2. But as we'll see, thereĪre other ways to do it. Illustrated definition of Geometric Sequence: A sequence made by multiplying by the same value each time. So this is one way to essentially define or create a function which generates this sequence. And obviously, if you startedĪt n equals negative one, then you're gonna get a different We want to generate the sequence that I originally wrote down. Zero, if we started at one, then this would be theįirst term in the sequences, which is not what we want. Then you're not going to get one of the terms in the sequence. If you tried to put a 1.5 year or something like that, The recursive definition for the geometric sequence with initial term a and common ratio r is an an r a0 a. However, the ratio between successive terms is constant. To specify that domain, where n is an integer and n is greater than or equal to zero. This is not arithmetic because the difference between terms is not constant. ![]() And it's equal to six times two, six times two to the n, where n starts at zero, and then it keeps incrementing by one. We could define a function, call it a of n, where n is referring to our index or which term in the sequence. And so one way to view this is if you view this as the zeroth term. This right over here is six times two to the third. So it looks like each term is six times a power of two. Multiple function definitions that could create the sequence. And then if that's the first term, the second term is now a 12, then a 24, then a 48, and so on and so forth. So let's just start withĪn example sequence. This constant is called the common difference. But we're going to focus on how we can generate the same sequence with different functions An arithmetic sequence is a sequence that has the property that the difference between any two consecutive terms is a constant. If you don't know what a sequence is, I encourage you to review Of this video is going to be on sequences, which you have hopefully already seen.
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