![]() The topics of this chapter will be briefly covered as they come up. Read as much as possible from this as soon as you can. Finding $\gcd(a,b)$ by brute force and prime factorization Uniqueness of representation $b=aq+r$, where $0\leq r\lt a$ Generalization to $a + (a+s) + (a+2s) + \ldots + b=\fracc_0$ Counting: $1+2+\ldots+(n-1)$ is the number of pairs. Counting: Euler's formula for planar graphs $v-e+f=2$. The idea that counting reveals patterns and determines complexity of things we are dealing with. The sum $1+2+\ldots+n=n(n+1)/2$ using a geometric proof. Example questions and settings that people study and how they relate to above. Big ideas: combinatorics, number theory, proofs, set/relations/functions, graph theory. The general topics one might encounter in Discrete Math. Each recitation instructor will use these as guide to cover what is appropriate given their recitation schedule.Īctual class notes for Spring 2024 (current offering, based on older lectures here) Recitations will be added here on a regular basis. Recitations (Recitation instructors: Shayan, Arezoo, Daniel, Anthony) (No more Navigate approintments, just walk in to Dolciani Math Learning Center on the 7th floor of East building) Tutoring Schedule (mostly done by undergraduate TAs) The team consists of 16 people and one Guinea pig. Any electronic document related to my courses (including pictures of documents you take yourself and any other sort of reproduction) that is accessible online must be on this site under Instructional Team Really ANY website, including within the Hunter College domain. It is ILLEGAL to distribute these documents or post them on ANY third party websites, such as coursehero, chegg, coursicle, studocu, studyblue, brainspace, kahoot, quizizz, reply, quizlet, tutor, stuvia, discord, youtube, facebook, instagram, X (twitter), etc. These document are the property of the author. You were trying to find say, the 40th term.Extremely Important: All the documents posted on this website, including notes, lectures, and homework assignments, are subject to copyright. Up with an explicit formula once we know the initial term, and we know the common ratio, this would be way easier, if Sure this second method, right over here where we'd come You might be a little bit,Ī toss up on which method you want to use, but for So this is equal to negative 1/8, times two to the third power. Is equal to negative 1/8, times two to the four, minus one. Using this explicit formula, we could say a sub four, So we want to find theįourth term in the sequence, we could just say well, We're going to take our initial term, and multiply it by two, once. Based on this formula, a sub two would be negative 1/8, times two to the two minus one. A sub one, based on this formula, a sub one would be negative 1/8, times two to the one minus one. We're going to multiply itīy two, i minus one times. We could explicitly write it as a sub i is going to be equal to our So we could explicitly, this is a recursive definitionįor our geometric series. ![]() We know each successive term is two times the term before it. Another way to think about it is, look, we have our initial term. Two times negative 1/2, which is going to beĮqual to negative one. Is equal to negative 2/4, or negative 1/2. It's going to be two times negative 1/8, which is equal to negative 1/4. Lucky for us, we know thatĪ sub one is negative 1/8. Then we go back to this formula again, and say a sub two is going Go and use this formula, is going to be equal A sub four is going to beĮqual to two times a sub three. We could say that a sub four, well that's going to be What is a sub four, theįourth term in the sequence? Pause the video, and see That is defined as being, so a sub i is going to be two Where the first term, a sub one is equal to negative 1/8, and then every term after Geometric sequence a sub i, is defined by the formula So I can reuse most of my equation from my simple example: a(i) = a(1) ∙ (2) ^ (i - 1) So for Sal's example, the terms are messier and we start out knowing only the first value and the multiplier, and the important information that it follows the rules for a geometric sequence.Įach term is 2 times the previous. If we want to find the 4th term, here is how we calculate it: In this simplified case I showed above, a(1) is 3 ![]() This is sometimes called the explicit formula, because you can generate any term if you know the first value and multiplier (common ratio). If you don't adjust the exponent by one, you will find terms that are in the wrong location. You can write a quick, general formula from this for all geometric sequences:įirst value x multiplier raised to number of the term, minus one ![]() If you have an original number of 3, your term numbers i would look like this top row. Another way to think of it is that every time you need a new term, you multiply by 2.
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