 So students from the homework number four Amy first of all it says show that this could be the first three teams of a Geometric series what they're really saying is is are the same from here to here to here How could I find are always? You remember I Think you thought it right and said it wrong. It's not the first one and divided It's any term divided by the one in front of it. So here's what I'm gonna do I'm gonna go eight x to the negative one-third divided by four x to the four-thirds That's our and this is great exponent review negative exponent Amy. I say to you elevator Instead of a pot a negative one-third on the top. I'm gonna move it to the bottom This is going to be first of all what's eight divided by four Amy on top This is gonna move to the bottom instead of four-thirds on the bottom. I think I'm gonna have five-thirds on the bottom Is that okay? Now to prove that this sequence is geometric I also need to show that this gives me look up that this is going to give me the same r If it is then I know oh you're multiplying by the same r over number. It's geometric So I'm gonna go 16 x to the negative two divided by eight x to the negative one-third Let's see if that also gives me two over x to the five-thirds Well, 16 over eight. I got a two Negative exponent Amy remember elevator trick. I'm gonna move it to the bottom and a negative exponent down here I'm gonna move it to the top Okay What's my denominator up here? I'd like to write two as a fraction over three. I think two is the same as Six over three is it not How many x is on top one-third how many x is on the bottom six-thirds how many left and where oh Are these two ours actually the same? Okay, so it is a geometric series. That's step one. Oh and if x equals eight So r is going to be two all over eight to the five-thirds Okay By the way, this is meant to be a tough question curveball. So like I'm glad you asked this Two divided by Eight to the power of I better put that five-thirds in brackets because there's two numbers in the exponent and anytime there's two number brackets I guess if x equals eight r is one sixteenth Is that a fraction between negative one and positive one? Does the sum to infinity exist then will this convert yes? Oh Let's find it now one minus r is gonna be one minus one over sixteen What's a what's the first term just read it to me? Oh But what did they say x was in this part B? Amy what did they say x was a okay? So I guess a is gonna be four times eight to the power of Four-thirds. Oh a is 64. Can you take it from there? That's gonna be the infinite stuff. That would be how they could say Okay, so much of this is straight plug-and-jug. What kind of curveball can we throw at them? And they love to with series and sequences do algebraic ones Whoo. Yeah, what's x? There's my first term put an eight in for the x there Let's write calculator Is that okay? Right, that's the sequence that it's got x's but they can tell me whatever x value They want me to try because hey let x be two they picked eight by the way because you take the cube root of eight Can you see all the fractions are thirds and thirds was the same as cube reading? So they probably wouldn't have used a two but an eight would certainly work Any others then Ladies and gentlemen, I think it's time that we started the last lesson of the course Yes, we're there at long last Can you please turn to lesson four Which is page 191 page 191 Natalia you with me you sure so you can put whatever you're looking at away Otherwise it will be with me for quite some time. Okay. It's called sigma notation and What this is is a short-hand short cut For a lot of information put your pencils down and look up Supposing I wanted you to do this Supposing I wanted to give you the following instructions starting with a five screen still frozen Sorry, I thought I did hit that I guess the remote's dying Supposing I wanted to give you the following instructions starting with the five Using a common ratio ratio of two add together the first 16 terms That's way too much writing for those instructions. Dina if you've learned one thing in math You've learned that we like short-hand notation All right Here is what sigma notation looks like the first thing that I would do is I would say Doug my friend They want me to add together. There is a symbol in math that tells me to add together and it's this I Pulled it out in physics a few times and you may have seen other courses because in English We've treated that as the sum of now actually I write it like this in type font. It's actually this And that's what it'll appear in our type written notes But when you're handwriting it it's a capital M on its side if you're really gonna be in a rush It's sort of a E meet Zed Sort of that symbol there tells any math nerd. You're gonna add some numbers together Okay Starting with the what five What's our here to to the M minus one does that part look a little bit familiar? Can you see a r to the n minus one? Okay How would I tell the person to add together the first 16 terms? This is the second part of sigma notation. I would say starting with n equals 1 Stop when you get to 16 That very concise notation right there tells a math nerd. Hey Plug in a 1 to this equation Plug in a 2 plug in a 3 plug in a 4 plug in a 5 stop when you get to 16 and add up all those numbers That's all sigma notation is right. It's a shorthand You have to memorize what means what but I always show this first now. We're gonna jump into the particular lesson So lesson 4 consider this 2 plus 4 plus 8 plus 16 plus 32 plus 64 plus 128 plus 256 That's a lot of writing Is there a shorter way to write this in this lesson? We're going to introduce a new notation which enables us to write the series in a very abbreviated form It's called sigma notation in the Greek alphabet the letter that thing Sigma which is where our capital letter s comes from That part of it looks like an s over the years. They stopped doing that and it became a capital letter s If you're wondering is used to represent the sum of a series The example below Represents 2 plus 4 plus 8 plus 16 plus 32 plus 8 represent this. Let's see. Here's what this says Here is my equation that I'm going to evaluate now This one is not in the form a r to the n minus one That's okay, and they're using a k you can use any letter that you want to but the sigma mat means I'm gonna add up each term and K starts with 1 means to write this out. I would put a 1 there. What's 2 to the 1? 2 and Then I would just keep increasing this number Stopping when I get to 8 so I would have what's 2 to the 2? Or what's 2 to the 3 8 dot dot dot? What's 2 to the 8? 256 and you can see oh it does write that out, but way shorter You can put any kind of an equation here that you want Usually we'll try and do a geometric series because that's this unit You read this as the sum of 2 to the k from 1 to 8 or The sum from 1 to 8 of 2 to the k We call this the upper limit We call this the lower limit and here's the general term Note the tally K is a natural number. You're going up by ones not by decimals, but the variable k is arbitrary They'll often use k or n or i i for index and For number and k for constant spelled with the k for some stupid reason But you can use whatever letter you want Okay So it says write this series in expanded form and determine the sum actually we're gonna do a different one. Are you ready? Do your little summation sign k equals 3 To 6 5 bracket 3 To the n minus 2 by the way 3 to 6 how many terms do you think there's going to be in this series? David 4 Not 3 Let's see if you're right. How many terms do you think there is you know what if there's only 4? I'm not gonna get fancy. I'm just gonna write them out and add them up on my calculator So what's the first term? I would put a 3 right there Dina what's 3 take away 2? What's 3 to the 1? 3 times 5 the first term is 15 I Put a plus sign because that's what this symbol tells me to do the sigma mean separate with plus signs What comes after 3? 4 put a 4 right there Matt. What's 4 take away 2? 3 squared times 5 the second term is 45 plus Put a 5 right there Adam, what's 5 take away 2? 3 to the 3rd is times 5 135 I think yes By the way, can you see a pattern in the numbers here too? What are we multiplying each term by it seems? 3 it seems so I'm willing to bet the next term is gonna be 135 times 3 That's gonna get me when a 6 6 take away 2 is 4 I would go to my calculator 3 to the 4th times 5 The final term is 405 David how many numbers are there? Yeah, you know how many numbers there always are This minus that plus 1 What's the answer? I don't know 601 600 even oh wow That's sigma notation When you see it you look at the initial index you put that in reverence as a variable and you keep Incrementing by 1 writing out each term separated by plus signs now Jen If instead don't write this down they had done that First of all how many terms would there be? not 13 14 you know what I would do I would write out the first couple Hey, that's a R is 3 and I could use my s formula from two days ago as a shortcut But you know what if there's five or less I'll just write them out and add them up because that way I know I'm doing it right next page Okay Write the series represented by this in expanded form and determine the sum. Okay What's my index? K going to be the first time k equals Five put a five right there. What's five plus one? Six the first term is going to be five times two to the sixth 320 and What I usually tend to do Brett just so I can keep track is above it I'll write the index and just underline it. That's when K is five Plus Brett what comes after five? So the second term is going to be putting a six right here. What's six plus one? It's going to be five times two to the seventh You know what the second term is and you said that's when K was six I'll also do K is seven I'll also do K is eight. How do I know to stop when K is eight Elizabeth because that's the upper bound That's what this notation means K is seven is going to give me Five times two to the eight 1280 and K is eight is going to give me five times two to the nine to the nine 2560 what is the expanded sum when I add them up plus 1280 Plus 640 Plus 320 4,800 yes, how many terms are there in this series? four How can you find the number of terms? It's always the upper limit Take away the lower limit plus one That's a great multiple-choice question by the way it shows up quite often Amrit They'll give you something like this and they'll say how many terms you know what they'd have for their answers three For the correct answer eight or five Because some kids just pick a number that shows up anywhere in the question Let's see if we can write a series in sigma notation Draw the little funky boom-boom-boom. I really should draw it in here There how many terms are there count? six Is there? So let's go from let's use K just for the heck of it K equals one to six Does that mean there's gonna be six terms? What's six take away one? Plus one is bigger minus smaller plus one so it works if I have a choice but tally I like to start at one although you could start with any number you wanted to you just get a different equation Inside the sigma than I did Take a look at this sequence though. This is a special sequence. This is geometric What's our? How can I always find our? any term divided by The one in front of it. How do I divide by a fraction? Multiplied by the reciprocal. I heard it back there. What do you say R was? Oh? Keep going R is gonna be three over two times one-third because dividing by three is the same as timesing by one-third Do the threes cancel? Yeah, R is a half. What's a? three Here is the shortcut for the geometric sigma notation Which depends on look up remember this Remember that two classes lessons ago three lessons ago It still works That's the easiest way to set up a Series and write it in sigma notation use the a r to the n minus one as your template But Kelvin to do that make sure you start from one and go to six So don't write this down ten plus 40 plus 160 plus 640 plus One more 6560 is that times by what's our here for? how many terms the sum from n equals 1 to 5 I'll use n just for the heck of it of a Bracket R to the n minus one. That's that written in sigma notation for me How can I make sure that n is equal to one? I? Just made that one up. I could write this a different way. I Could go if I wanted to Kelvin for up. I could go if I wanted to from n equals Two to six what six take away two Four plus one So five terms, but you know what for that to work. I think I'd need an n minus two right there Different equation works though I could go from like 17 to Something you know what if you have a choice start with one and it fits our normal template already example for Now an example for are they starting with one? No zero. I'll be careful and An example for Jordan an example for How many terms are there in example for? Yes with me Yes, I don't know what you're doing with me Brett. How many terms are there an example for it's a trick question Caste goes from zero to what ah This is an infinite sum like we did last day first of all When they give me an infinite sum when they give me any sigma You know what I always do I write out the first three terms because I can always spot the pattern from that So you're ready. I'm just gonna write out now Brett when K is zero. What's one-third to the zero power? What's anything to the zero power? one times five Five that's the first term Plus When K is one I'd have one-third times five Yes, by the way, I'm pretty sure this is gonna be our almost always what's inside the exponent is our Plus when K is two I'd have one-ninth Right one-third squared Which would give me five over nine when I multiply it. I'm just gonna go like that and see if I can spot the pattern first question What's our you can find it Nick two ways Any term divided by the one in front of it, but I think you're gonna find our is one-third if You're doing sigma notation and there's a number and the index is the exponent That's your art. Here's my question Nick. Is this a fraction between negative one and positive one? Will this infinite sum exist? Moving over one Vitaly, is this a fraction between negative one and positive one? Will this infinite sum exist yes or no and how do I know instantly without having to think about it? Assuming I've done the homework from last day What if it was this? What was our test to see if a sum to infinity which is what this is? Existed how did we know instantly? The ratio what about the ratio? Had to be fraction between zero and one so I'm coming back to you Nick my friend. What's our Does the infinite sum exist? Why yes? So I can continue What's a that's why I wrote out the first three terms Jen so I could spot a easier How can I find an infinite sum last day the in the answer to this guy is? Going to be five all over Dan in your head. What's one minus one-third? Yeah How do I divide by a fraction flip it and multiply? I think it's going to be 15 over 2 or 7.5 So I can take sigma notation Dylan and I can crank an infinite sum from last day out of there, too I just have to be careful I always write out the first three terms just to make sure I know what a is a is usually the number in front not always Usually or is usually the exponent number not always and it takes all of what? 30 seconds to plug K in three times and get those three and then I know Dina is scrunching her face looking at example five and The reason Dina is scrunching her face looking at example five is example five is actually from the Alberta curriculum So the very last question of this year We're going to take our pencils or our pens and we're gonna go Number one you know what Let's do number one together right now Number one says state the number of terms how many terms in this series right here six How many terms in this series right here? It's going to be 32 take away 17 plus one 32 take away 17 is 15 plus one 16 Mmm. I like question C. I like question C. I like question C How many terms in this series here? It's gonna be a plus nine take away a Plus one What is that really Matt? 10 see it the ace canceled and by the way that looks so terrifying when kids see that on the provincial for the first time The number of kids that pan no, it's always the top minus the bottom plus one and in this case Oh the ace canceled. Yeah 10 To a but cross out the middle look cross out that to a part one and three Write it out an expanded form. How many terms will there be six? And then Adam up C and D Logs. Yes logs number three Says write each series in summation notation Which rhymes conveniently I'll go A and C I'm gonna skip four skip five Skip seven nine is nice eight is nice Ten is nice by the way eight and ten. It's a little bit of review of logarithms Are we almost at the end of the course? So I'm sticking some of that in now. Okay What else is your homework a short lesson? Okay, what else is your homework? I think you can also now we're done the course if you finish this homework Make sure you're caught up through the series and sequences homework This is also a good chance for you to do a little bit of review Maybe go through the written section of one of the five provincial exams that I or provincial exams that I gave you the other day What is your series and sequences unit test? Tuesday next class Next time I see you The test the multiple choice will be Series and sequences eight or ten questions and all likelihood the written is gonna have are you ready? Are you ready a quadratic trick? Not an identity I'll do identities later transformations Either reciprocal or Something with expansions compressions reflections and slides with all of them Remember the order? Okay, so that's transformations That's trig what am I gonna do for logarithms? I think for logarithms It's going to be an exponential equation Something like this and I'm making this one up something like eight equals three bracket five to the two x plus one Okay Divide by three to get the exponent by itself and then take the log of both sides and you'll end up having to Muck around with that. Okay. I know it means you need to do some review. Okay That was transformations that was trig that was logs Yeah, I'll put one combinatorics probably a committee question Ten boys five girls Picking five people how many ways to get exactly three boys and two girls how many ways to get at least three boys of two girls That'll be the written. I think I answered that already. All right So that'll kind of jog your memory. I really think it'd be worthwhile sometime this weekend To navigate to the pit math to where it says Principles of math 12 links to where it says click here for video tutorials like stuff And if you shake you got a couple units watch some of the tutorials What I'm really trying to do is give you a mini mock on the written So that the real thing isn't a shock. I have one more thing I have to do Before I turn you loose