Unit
Sequences and Series
University of Michigan
Runs his own tutoring company
Carl taught upper-level math in several schools and currently runs his own tutoring company. He bets that no one can beat his love for intensive outdoor activities!
University of Michigan
Runs his own tutoring company
Carl taught upper-level math in several schools and currently runs his own tutoring company. He bets that no one can beat his love for intensive outdoor activities!
Lists of numbers, both finite and infinite, that follow certain rules are called sequences. This introduction to sequences covers the definition of a sequence and how to identify a rule. There are specific sequences that have their own formulas and methods for finding the value of terms, such as arithmetic and geometric sequences. Series are an important concept that come from sequences.
We're now going to talk about sequences and basically what sequences are is a row of numbers some with some sort of pattern amongst them, okay. So and how you there's basically some relationship between a number and the one before it, okay? It could be adding a number to it, it could be multiplying. There could just be some different pattern to dedicate this whole sequence of numbers, okay?
So behind me I have 3 different examples of sequences, okay? Our first one is 2, 5, 8, 11 and hopefully what you see is that in order to get from one number to the next, you're just adding 3. Okay? So you add 3 to get to 5, add 3 to get to 8, 11. The next number would be 14, 17 and so on. Okay.
The next one goes 3, 9, 27 and 81, there's a couple of ways we could break this one down. Okay? The way that I intended it to be done is powers of 3, okay. So we have 3 to the first, 3 to the second, 3 to the third, 3 to the fourth. So on and so forth. So the next term would be 3 to the fifth. But another way you can look at it is you're taking the previous term and multiplying it by 3 to get the next. 3 times 3 is 9, times 3 is 27 times 3 is 81, so on and so forth, okay?
The last one is very similar to the one above it. But instead of powers we're just dealing with strictly multiplication. So you take 3 multiply it by 2 to get 6, by 2 to get 12, 2 to get 24 so on and so fourth, okay?
So these examples all our numbers are always getting bigger but there's no reason our numbers couldn't be getting smaller as well. Okay? So here I add a 3, I could have just as easily subtracted 3. Here we multiply it by 2, I could have just easily divided by 2. As long as we do that consistently what we're getting is a sequence of numbers.