 Hello and welcome to a screencast about sigma notation. So sigma is a Greek letter, and as you can see here, you've got a number at the bottom along with the variable, and then typically a number, maybe another variable, up here on the top. And then here, we've got a function, just like we've always done functions before. And over here, I put a box and it kind of explains what's going on. So the bottom number is your starting index. The top number is your ending index. We're calling the sigma s sub n, that just means sum. And then this a sub i is your sequence formula, so it's also your function. So the main thing to remember with sigma is that it just means you're going to be summing up a bunch of different terms. Now what terms you're summing up is going to depend on the indexes down here below, or the indices, the one below and then the one above. OK, so an example of this one, this direction, say write the sum longhand and then find its value. So this sum goes from 1 to 4 in our variable. In this case, it's going to be i. You can use n's. You can technically use x's, although that's not usually the way you're going to be doing them with these. Usually n's, sometimes j's or k's is a typical standard notation for these. Anyway, so our formula for this one's going to be 3 times 2 to the exponent of i. So all this means is we are just going to plug in. Basically, you're going to start plugging in numbers from 1 up until 4. And these indices are always whole numbers. Sometimes they'll be negative, but typically they'll usually be positive. Sometimes they may start at 0, but oftentimes they'll start at 1. They could also start at 15, so it really doesn't matter as far as that goes. But whatever number you're starting with, you're going to use all the whole numbers between the bottom number and the top number. OK, so for this one, we're going to plug in 1 first. So 3 times 2 to the first power. And then we're going to plug in 2 to the second power. And then we're going to plug in 3 to the third power. And then we're going to plug in 4 to the fourth power. So that's writing the sum longhand. So that's why this notation is so nice, because then you don't have to write out all these different terms that you've got here. OK, so then we can just crunch out this value. I guess I probably should have done this ahead of time, but that's OK. So we've got 2 to the first is 2 times 3 gives me 6. 2 squared is 4 times 3 gives me 12. 2 cubed gives me 8 times 3 is 24. And then 2 to the fourth is 16 times 3 gives me 48. You'll also notice that these numbers, I could have actually figured them out much easier too, because you notice that every time they're increasing, they're doubling basically. So 6 doubled gives me 12. Double 12 gives me 24. Double 24 gives me 48. Just for the record. So anyway, if we combine these, let's see. So if we combine 12 and 48, that'll give me 60. And if I combine 6 and 24, that'll give me 30. So I believe my grand total for this sum is 90. And hopefully my mental math and my arithmetic there is correct. All right, so now the second example here is going the other direction. So in this case, I gave you some numbers that were summing up together. And I want you to turn around and write it then in sigma notation. So this one gets a little bit trickier, but it's kind of like what I was babbling about in this last problem. You want to look for a pattern. So what do you notice is happening with these numbers? So as you go from 2 to 5, is there something that's happening that the same thing happens that you can go from 5 to 8? Or 8 to 11? And the 23, that's not really not going to help us. So hopefully, you notice that every time you're going to be adding a 3. So in that case, thinking back to your algebra, that's a linear idea. So basically, for this one, this has a slope of 3. So let me write that down, slope of 3. OK, now our sigma notation. So as I said before, it depends on where you want to start. So we did eyes on the last one. So let's go ahead and do eyes on this one too. And we'll just start it at 1, again, since this one started at 1. But that's going to definitely change what your function looks like, because what these numbers mean is if I plug 1 into my function, it's going to have to spit out 2. If I plug 2 into my function, it's going to have to spit out 5. So this one's going to correspond to a 1, a 2, a 3, and then a 4. But let's say I started my indices here at 0, then this one would be 0, 1, 2, 3. So it may be off by a little bit, just depending on what you're doing. But just so you know, it's all the same, it's just going to look different in your sigma notation. So basically, there's more than one correct answer for these, what I'm trying to say. OK, so if I'm going to let my first one be 1, then I need to think of a function. So I know my slope is going to be 3i. I already said it was linear, so it's going to have to be plus b or minus b or something. So what when I plug 1 into this, is it going to spit out 2? Well, so if I do 3 times 1, that gives me 3. So I'll have to do minus 1, then, is going to be my function. And I typically put these in parentheses, just so you're not confused about what you're summing. Because if you don't put that, then you may think, oh, I'm just summing the 3i, and then I'm going to subtract 1. So it's always just a good idea just to be clear about what's going on. OK, so then let's double check some of my other values too. So if I plug 2 into this function, will it give me my 5? Yep, if I plug 3 into my function, will it give me my 8? Yep. OK, so now I've got to figure out, what am I going to plug into my function to get out my last value here of 23? And then that's going to correspond to what my ending index is going to be, my ending index. So we could just do this by algebra. So let's do 3i minus 1 is going to equal 23. So then 3i is going to be 24. So i is going to be 8. So I know I'm going to stop my sigma notation here at 8. And sorry, my sigma I realized was not exactly very flat at the top. But yeah, you get practice drawing these. Let me try another one. Just like when you guys learn how to do infinities and all that fun stuff. Ooh, that one's better. OK, there we go. So anyway, there's my sum and sigma notation. And I believe that does it for today. Thank you for watching.