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8.1Lesson #1 2019

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this is a partner in this is a new chapter chapter 8 lesson 1 introduction to sequences series and summation notation Market to do this okay if you collect marbles or DeWalt is a collection of numbers instead of Mom isn't all that is ordered in the key word here is water what's the numbers that is bordered 246 is not the same the sequence 64 so not the same as 6 to 4 because the order matters set the sequence 3 6 9 12 never ends it is called and infinite secrets the sequence 3 6 9 12 ends therefore it is finite sequence and number in a sequence is called a term each term is separated by commas so using 3 6 9 12 an example the first term is obviously three and the fourth term one-two-three-four is obviously 12 besides writing that each term how can sequences be Rewritten or be written it takes a while to write if you have 20 terms is he going to have to write them all out so therefore it has to be a shortcut on a sequence can be written using an equation which is called the rule of the nth term what is the rule of the nth term and it's denoted by a subscript end where the end is the term number show the sequence 3 6 9 12 can be written as a subscript and equals 3 and we will talk about how to come up with those rules little bit later how do we find the first four terms of the given sequence try to find the first four terms the given sequence what you need to do is you first of all need to substitute the term number and equals 1 into the equation then you sent and repeat 4 m equals two and so on let's go ahead and do that so the first term would be -3 to the first and make sure you using parentheses step first term is -3 and then you would fight in the to substitute in at 2 with and so on and then we will write this listed as a list send a letter answers  390 27 + 81 that's how we would write the first four terms so you show your work and then you write it as a list so how can a  offender which we know is the rule how can that be found for the nth term I want to do is we want to look for good color 4 pattern later on we can be learning a better way of doing this but right now we're just going to be looking for a pattern here in a great hint we are going to write to term number above each turn that's really going to help us fine I never going to rewrite the sequence possibly using fractions or not but this possibility and then you're going to guess and check you need to put on your thinking cap for this one because these are not really that easy yet to do and you'll probably be having coming off your brain yes truly a well okay so put that thinking cap on and let's get ready to come up with some good answers I just have and the hint that you want to ask yourself is how we cite evidence is how do I get the term from its in so how do I get the term from its turn number which is in so from its number I must remember so how do I get the term from its turn its in which is turn number thinking cap relation that we could have fractions is how do we get that term from its term it's an which is to remember and we can write that term about it so that you think about it I want to get to fractions a little hint you want to find the rule for the numerator separately spell separately right from the rule of and this will if you will help you a lot try to do it all at once it's going to be confusing so I don't want the numerator separately from the road from the denominator and then put it all together so what is the rule for the nth term for each of these 22 write down what the end is so at this is my term number when is 1 2 3 4 5 so then I'm asked myself I get the one from this one here Jessica the end is possibility if it's just an end then this one would be at 2 so that's not going to work 16 right here how can I get the two how can I get the eights in the two could do two times for which would be 8 so that I'm going to try that idea with the three and you just watching right now three times 4 is twelve that's not going to work another way that I can get the eighth from the two yeah I think you just said cute didn't you so let's try that too cute is 8 1 cubed is one 3 cute is 27 for cute is 64 V Cube is 125 I think you got it there for end cubed will give us our and turn so therefore our will is in cubed great job everyone he's just going to play around with it by squaring try multiplying try to fighting type Powers Etc are there so many different ways to do it I'm faces yeah you just going to play around with it and see what you come up with when you do your homework next one the hint here is a numerator or denominator after you write what the ends are write this down so this is going to be separate numerator denominator rules let's start with an easy one is easier to find a new way to hear what is a meter correct the seven the denominator will always be 7 no matter what the end is that was easy so now I'm going to focus on these how am I going to get from the end to the term itself so how do I get from 122 from two to three sons 44556 I think guys got it you were just going to to add one and so then that's our rule up there is and plus one because it didn't do so well on homework I can tell we are done with sequences and series does positive you need ketchup what is a series well looking at this down here looks pretty much like a sequence it would be a sequence. Changers so what did I replace the commas with in order to get a series correct iplus so there for it is day answer to a bunch of pluses and especially expensive classes would be some so it is the sum of the terms of a sequence is there a series 3 plus + 9 + 12 is finite it ends there is no... But the next one has... Which means it is infinite next one is 2 - 4 + 6 - 8 + serious why or why not yes it is one and I think you did get the answer the reasoning for it I can rewrite it right by changing the subtraction to adding the opposite so therefore instead of leaving is 2 - 4 + 6 - 8 + 10 for someone that didn't understand what was going on you would just rewrite it as 2 + -4 + + -8 + 10 + now that looks like a serious because there's pluses so therefore if you see a plus or minus alternating like something like that can you can tell alternating glasses and minuses and there's a pattern then it will be a series what is a special way to write a series now we're talking about a special way and this is called summation notation which is also known as Sigma notation we're going to talk about the parts of rotation make myself a line here so I know where not to pass okay so summation notation let's go red and raw just make sure you click on in the middle because you're going to be writing down what parts are in the book it does is ISO lines on there but I usually can't write very well so if you just like Annie that's okay and then this part right here is the rule and up here is a number we're going to call it something and over here we have a letter no better that's going to equal another number I think we have all Shepherd so let's talk about what each of these parts so this letter right here is called the end and we usually use an n a k or an i that's what we usually use I'm going to use a lot of end the book Holes is mostly and sometimes Kay's very seldom eyes and this let number right here is called the lower and that's going to be the number that the term member will start at and then up here is going to be is called the upper limit and that's the number term that you can do it and then over here we're going to put some examples the first example when I write these is kind of writing like with an E and has an equal one and it's going to go to buy so turn one two three four five in French season about classes there's the first example another example sometimes you might start at the numberless use a different letter to so they might start with the third term of the series that they want you to find a partial sum and then up here than it might go to Infinity and I have a rule what's the price is right now but you don't have to have these are my examples and this part right here is what you guys are going to need to know you don't need to put a box on that unless you want to and this weird looking e it's called two Sigma and that's an uppercase Greek letter and a mask some that little symbol is called segments uppercase Greek letter and if that means some so how do you read it I know how to read so basically what happens if I looked at and I can read it and it says you probably wouldn't be able to yet but if you look at you go it's that you with the end is down there with a 1 equal to it and then up high as the 8th and the the rules to end so maybe I will understand what you're saying but it's better to learn how to read that be like in the sentence the dog ate the dog food correct way but you could say food dog ate and it's nothing so we're going to learn how to read this so you're going to say this as the sum from and equals 1 and you say to eight infinity turn vanity the rule which is two then so you say the song from n equals 1/2 insanity of 2 what is the song the song from n equals 1 to Infinity of to the end in general you say the sounds from whichever the whatever the index you say the songs from the index if I send k i a song from DMX and equals the lower limit to the upper limit base of skull pain Southern General that's what are you read it and you will be responsible for knowing this it doesn't hurt to practice so what you might want to turn this off and practice reading the ones up here in the purple the song from equals 125 of 2 and 3 S1 the song from n equals 125 of 2 and plus one that was this one up here the second one is the sum from CaCO3 to Insanity at 3 the sum of CaCO3 to Insanity of 3n he might want to practice those what is the summation notation for each of the series below so in order to do these you have to find a wolf or a subscript in and so we're going to do what we did to find the rule for a sequence and then you just put in summation notation and find upper and lower limits and put those in there so you can just follow the rules. over here the first of all I'm going to write in and then this is one two three four five six and I'm going to say what am I going to do to the end to get for when I going to do to the two to get 8 when am I going to do to the 3/2 well what am I going to do to the 4 to get 16 you probably figured out by now you going to multiply it by or so take away those question marks you try One X for that gives me 4 2 * 483 * 4 gives me 12 four times for Gizzy 16 5 * 4 equals 25 * 6 equals 30 + 20 + x 4. my rule so if this is my role then I do summations notation I'm going to use an end and eat this is my first term quotes from when and how many turns do I have 1 2 3 4 5 6 6 times showing but that's infinite so therefore because internet up here I'm going to put Infinity and then my role is 4 in I find a rule set it up looking like this and then I find it from if you notice says it's just a concise way to write a series so this right here is the same as that up there the next one let's figure out this role the role and equals 1 2 3 4 5 6 looks like for the rule the numerator is always going to be one how do I get from 122 to +24-328-416-5232 and 6064 multiplying by 2 isn't cutting it squaring it isn't cutting it cubing it isn't cutting it looking right here so let's try some Powers I see that these are powers of 2 so let's try that to to the first good to the second to the 3rd to the 4th to the 5th to the 6th so that looks like that's going to work he play around with them so they my denominator is 2 to the nth so when I put all together and shoot again and goes from the first time without the 1 2 1 2 3 which ones we have one 1 term 2 4 5 6 and no... That means that it is only two 6 is finite no... Finite and then my role is 1 / 2 and tell her and you can leave it that way or you could have also done 1/2 raise the nth power to if you thought that was a rule quadratini to let's move on and finish this so what does it mean to expand a series that is in summation notation using the rule you can find all the terms list all the terms of the series starting with d Laura limit value and ending with the upper limit value infant infinite series you just I just want you to the first five terms and then put your... So now I'm doing the summation notation and it's called expanding going back to the list expanding is basically listing basically what it is so you going to find your first turn + 1/1 squared equals 2/1 second term cuz we're going to 4 the third term and then your fourth term opcity organized a little bit better she can see that you can probably follow that a little bit better with it being spaced like that so then expansion would equal to exactly the same thing too and it's a sound remember this means some SO2 + 3/4 + 4/9 + 5/16 and then for the second one are we doing is expanding its called expansion so then I'm going to do the first term it's going to Infinity so I'm only going to do the first five terms 2 + 1 which is 3 a subscript 2 2 2 x 2 + 1 which is 5 I just seven right around this time if you started writing these up here you're going to see a pattern emerging and what do you notice how am I getting to that next turn heading to so I think this may be a sign that says check it out and it is equal to 9 and so the next time would be 11 tried out the pattern and that is the case so you can play around with that if you want to see the pattern and follow the pattern is also powder on a tooth that you could follow the next one would be 6/25 and lastly set up for Canik Spanish series and we're going to find the sum of series at the end let's find the sum what we're going to do is freaking expand Spanish list make sure that your listing all the terms list of terms of the Sun it sounds so we're going to listen Mall expanded and then add them together so this evening do these and I had so one- one + 3 is 2 II -2 + 3 is 1 and then the third one if we do in their head what's going on in your head the fourth one and then 1 we're doing our head I see that only go to 5 so you see five here five here we started with the one so now my son is to be equal to zero and then the next one is a different song set zero so now you don't have zero terms and I think we're going to do is go ahead and this change it to 1 we're not going to go into the zoo turn jet so let's start that with the one so I'm going to put it in my first term I remember that just means to do that first it means negative One X to the first that equals -2 and then you put it to you in there which means -1 * 2 squared I need to pass negative or you can just put subtraction put a 3 in there just means Neither One X to the third which is -8 and then you find your son she's going to be negative 14 I hope you noticed that I change this and I'm lost start with a Wendy's diss when you definitely can put in your head 5 * 1 is 505 * 2 is 1005 * 3 is 15 V * 4 is 25 * 5 is 25 and it goes on and on and can you find the sum if the numbers are increasing no there's no some that's because at 10th in it Everlasting wasn't easy way to differentiate between a sequence and a series sew sequins uses something and the series uses something else so what does a sequence to use Camas what is a series use or minuses and we are done

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8.1Lesson #1 2019

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