CIE A LEVEL Maths FULL SUMMARY Topic by Topic Paper 1: SERIES

and welcome to flip Academy today we're going to be looking at the series section of paper one for the cie a level maths so if we have a look here uh the first thing you need to be able to do is use binomial expansion and in this case it's going to be a plus b the power of n where N is a positive integers keep in mind when we don't have positive integers and we have like different sets then binomial expansion does change slightly so this is just for very standard normal brackets so for example if you had something like 3 plus x to the power of 20 right because expanding those brackets would take you a few decades so we have a special little way of doing it and that special little way is using something called the NCR button on your calculator so what we're going to do is I'm going to write out the formula for you so we have in general a plus b to the power of n is equal to First a to the N nice and easy and then all you're going to do is have the NCR button following that so you have n C and then position one and that's going to be B to the power of n so yeah a to the N minus 1. so all you're going to do here is as you go along you're reducing the power of a by one and increase in the power of B by 1 as well so then we're just going to keep going so that's that's here n c 2 a again subtract one so it's minus two and then B squared and then we can just keep going you can just keep doing it so just remember which numbers you are increasing and which ones you are decreasing like that and then B you keep in mind this isn't a bracket so sometimes what you might see is that people make it like a doubly thick C just to make it super clear that a is a c and not a bracket so the best way to do this is using your calculator so if we get our calculator to open I will show you exactly how to do this okay but in this case what might be good is if we actually write down the fact that we've got a is 2 here and B is minus and B is here as minus X okay and then obviously n is six so what are we looking for we're looking for X Squared and X cubed so the x squared term will be where B squared is here so we're going to be looking at this section here and this section here so what we're going to do is I'm going to write out x squared put a little colon and that's going to equal n which is 6. the C button again we can make it like slightly thicker to make it super clear it's not bracket and that's going to be position two and then a in this case is actually three so we're going to that's right it's two so put two n minus 2 would be four and then times B squared so again keep in mind B is actually the whole thing so it's minus X or squared okay and then that's going to be plus and then all we're going to do is look at the coefficient after we've done that so if we got our calculator back once more the NCR button sits right here it's the Divide sign so we put shift divide n always goes on the left and it never changes so six and then you put the two now here's the key thing okay you can't just write down extra numbers now you have to put the time sign okay you can't even use a bracket so just make sure you do that then we have two to the power of four I do like to put these in Brackets um but again you don't I mean I always put everything in Brackets just to make sure and then obviously I'm not going to put the X in but we're going to put minus 1 squared that's 240 so that term that whole third term is going to be 240 x squared so the coefficient is just 240. then if we look at X cubed I'll try and write it slightly neater this time again the 6 stays identical because it's n we've got these slightly thicker C that's going to be position number three we have two to the power of n minus 3 n is six so we have 2 cubed times minus X cubed and that's going to give us so again I'm not actually going to delete any of this I'm actually going to go through and just edit it so when you're doing the full binomial expansion I would recommend just you know changing it rather than rewriting the whole thing negative is minus 160 x cubed so again in this case the coefficient is -160 and that there is three marks which I think is pretty generous now it says hints so if you remember what I've said in all of the other videos whenever they used the word hence it means you have to use part A if you use part A and make your life a thousand times easier to find the coefficient of x cubed in the expansion three X plus one and bracket okay so here's the tricky bit okay this is something you have to remember when we multiply that when we expand this thing out the only X cubed terms are going to be when 3x minus 1 multiplies X cubed so x squared and when we have 3x minus 1 times x cubed because think about it when we have 3x times x squared we're going to get an X cubed term then we get minus x squared so we can ignore that then we're going to get 3x times x cubed which we ignore but then we have minus X cubed so what you need to remember here is that we are going to we have to include both of these terms if you notice we already have the coefficients so since we're only finding the coefficients all we need to do is actually find them so what we're going to do here is I want to replace this with 3x minus 1. times the x squared term which is 240x squared which equals 3 times 240 is 720 x squared x cubed even and then we have minus 240x squared now again I don't actually care about this term right I care about this one here then we have 3x minus 1 multiplied by the X cubed term which is minus 160. x cubed three times that would give me 480 so it'd be negative 480x cubed x to the 4 even minus so we actually plus 160x cubed because again we have a minus times a minus so we have these two parts here so the overall coefficient of x cubed is going to be 720 Plus 160. which is 880. okay because we're going to be gathering the like terms so the reason why this is important is because the only other way to do this if you don't know this trick is to actually do the four six term expansion here and then expand it by another bracket which will get you the right answer don't get me wrong but it's going to take you five times longer it's going to take you a hell of a long time um and it's tedious it's boring the only terms that will give you X cubed are when you expand the x squared term and when you expand the X cubed term so once you can kind of make that connection in your head you can shorten this question down to the two marks it should be okay the next thing is we need to be able to recognize arithmetic and geometric progressions so first of all we should talk about what the difference is between them so arithmetic just means that you're adding the same value each time so I say adding but keep in mind you could be adding a negative number so actually you're subtracting the same number each time so for example you might have one three five seven as you can see it's going up by two you can have ten six two so as you can see it's going down by four okay geometric progressions have a common ratio so this here is called a common difference here is called a common ratio so what that means is you are timesing or multiplying which also technically includes division right by same number each time okay and you just need to be able to recognize them so how can you recognize them because sometimes it can be a bit ambiguous well the best thing to do is if you have a random progression so let's say will be a good one to do um one for then a for example going on you can like for example it could be timesing by four each time right so the common ratio would be times in my four because one to four times and then times so one way that I like to look at it is if you can't see a common difference then it must be geometric but the best way to tell is you just simply do 16 over 4 and you do four over one and if you notice you're going to get the exact same value of 4. that means there's a common ratio because keep in mind that these aren't the only types of progressions that exist in the universe right they're also Fibonacci sequences Etc so you need to be aware that if you want to prove that it's geometric or arithmetic you need to actually work up the common difference or the common ratio if there is no common difference and no common ratio then it's neither an arithmetic sequence or a geometric sequence okay so you need to keep that firmly in the back of your head as you do these questions the next part is there's a ton of stuff here use the formula for the nth term and for the sum of the first n terms to solve problems so nth term for let's split this whole thing in two and arithmetic sequence is a plus n minus 1 D where a is the first term so in all of these uh this is going to be the first turn and is obviously what term number you're in D is the difference I'm going to use the letter R later which is the common ratio okay for geometric it's just a r to the N minus one so they both have that n minus 1 in them for the sum you're going to have for an arithmetic sequence over two D and for the geometric one this is going to come up a bit later but just for you to know now it's a to the r n minus 1. over R minus 1. okay so you need to be able to use those to solve problems and honestly these actually become quite easy because there's a very limited number of things they can ask you and once you kind of practice them a little bit you're going to get a lot better at them you just need to use the idea of convergence of a geometric progression and the formula for the sum to Infinity of a convergent geometric progression so what what is Convergence well convergence is that as you go along in a sequence it eventually converges so it gets closer and closer to a certain value so what do I mean by this well if r the magnitude of it is bigger than one then hopefully you can see that the terms will just keep getting bigger and bigger so I'm going to generate a sequence here right so let's say one equals and our answer this is going to be whatever term I am let's say the value of R is 2 right so we're doubling each time if I keep doubling right can you see the number just gets bigger and bigger and bigger and bigger this means it's a Divergent series okay now let's take a different example I'm going to take 10 as my first value and I'm going to times it by let's say 0.9 so we get 9 then we get 8.1 then we get 7.29 now let's keep going for a bit so 4.3 3.87 3.48 can you see that the numbers are getting closer and closer together all right so the idea here is that this is a convergent series because it's getting closer and closer to a certain value okay now if if a series is convergent it means you can do a sum to Infinity so what is the sum to Infinity well an SN is a sum of the first let's say 10 values or 20 values or 30 values now if my series is going towards one certain value then I should be able to add up an infinite number of terms and not get infinity because it's going to eventually get for example let's say if the series is approaching one eventually I'm just going to be continuously adding one right so then it's just going to converge to a point or if the um series tends to zero as I'm adding the numbers up and up again it's approaching zero which means I'm making less of a difference okay so you need to be able to use that now the sum to infinity and again this only works if this isn't true so you need to have your modulus of R is less than one because it needs to be able to go to a certain value normally zero is just a over 1 minus l nice and easy or alternatively the difference here is that this bottom part should be positive okay so just uh keep that in mind so if your value of R is different you're subtracting one okay so we have a bunch of uh practice questions for this so an arithmetic progression has first term seven so I like to actually write down all of the information they give me the end term is 84 and the three nth term is 245 so normally our nth term is this what they're telling me is that when n equals n right we have seven plus n minus one we don't know what D is is equal to 84. and then when n is equal to 3n we have 7 plus 3n minus one D equals 245. so I'm going to do now is I'm just going to subtract 7 from both sides for both of these guys okay so we have n minus 1 D equals 84 minus 7 which is 77 and then we have here 3n minus 1. D is 245. there are so many ways to solve this you can use substitution however you can use a clever use of elimination if I label these one and two if I divide the equations D actually goes away so we have n minus 1 . and the reason why I haven't written the D is obviously we have D divided by D which is just one so it disappears over 245. now again multiple ways to do this I prefer to kind of cross multiply these out so what I'm going to do is times through by three n minus 1. I multiply through by 245 at the same time so on the left hand side we're going to have 245 n minus 1. and on the right hand side we're going to have 77 3n minus 1. and then from here it's a linear equation you just need to expand and simplify I didn't actually do the subtraction there that should be 238. okay so we have 238 n minus 238 equals 77 times 3 is 231 so I'm going to bring the ends over to the left hand side and the numbers over to the right so we have 7n nice and easy equals minus 77 plus 238 would be 161. so n is just going to equal 23. it's again not too bad you just need to and again with these questions they're always going to give you enough information to actually answer the question right so all you need to do is write out all of the information in terms of equations and in terms of just substitutions like I did with a and normally the solution just then presents itself and it's quite okay next we have a question from the 2019 paper one so we have the third and fourth terms of a geometric progression of 48 and 32 respectively find the sum to Infinity of the progression okay so if you remember a here's what we need for the sum to Infinity so if we just quickly write that out that's going to be a over one minus r so the two things we need is we need a which I don't know and we need R which I also don't know so how can we work it out well we have two adjacent terms right here okay so what if I take uh the fourth term and divide it by the third I'll get R because what that will give me is what I times the fourth term by in order to get sorry what times the third times the third term by in order to get the fourth term and that is the common ratio it's the very definition of it so we're going to do is do 32 over 48. which if you divide it by 16 you get two thirds nice and easy bear in mind that um a lot of these questions can be calculator as well so don't worry too much about that but yeah okay so now we need to work out what a is we know what this is that's good we need to work out a is well again using the same logic n equals 4 gave us 32. n equals 3 was 48. so all I need to do is continue dividing by two-thirds okay and if you remember dividing by two thirds that'd be the same as times as my three over two so you times it by three and then half it and you just keep going from there so that'd be 72 and n equals 1 would be 108. okay so now we know both of those which means we can now just use our sum to Infinity all is well and good is equal to 108 over 1 minus two thirds so in other words it is 108 over one third so you're just doing 108 times three so that should be 324 right because 8 times 3 is 24 100 times 3 is 300 smush them together and you get your answer okay now here's where things get more fun this is where they apply it into certain situations which I always find a bit more interesting and this is where you can start picking up I mean in this case five marks and honestly you don't need five marks for this it's really not too bad so you don't need five minutes so if we read through the context two schemes of increasing the amount of household waste that is recycled each week scheme a increases it each month by 0.16 tons and B is to increase it by six percent each month of the amount recycled in the previous month so if you notice a is an arithmetic sequence because we're increasing by the same number every single month and B is a geometric another way to kind of look at this is if you have a percentage increase every unit of time or every time you do something it's always geometric right it's just like doing compound interest back in GCSE okay because a percentage is always going to be it's going to be different every year right like 100 increased by six percent is 106 but increase that by six percent is going to go up by more than six uh so hopefully that makes sense so if you have a percentage it's 99 of the time going to be geometric if you have a set amount it's obviously going to be arithmetic and all we need to do here is so they're going to operate it for 24 months the amount recycled in the first month is 2.5 tonnes so for both of these guys a is 2.4 oh sorry 2.5 and all we need to do is find the total amount of waste that would be recycled over the 24 month period and this is the kind of calculation that people do in the real world right so before they put into place this scheme right they can work out how much in total over a certain period of time will each one do so just as a bit of context for this short term right you can see the 0.16 is going to be bigger than six percent of 2.5 short term but if you're doing it over a period of 24 months it gets a bit closer long-term percentages always win compound interest right it always wins long term but the question is how long term are we talking is it five years 10 years 15 years who knows so these kind of calculations I mean I'd be surprised if you can look at this and actually work out which one will be better just by looking at it but um yeah so all we need to do is calculate it so the sum for a arithmetic sequence as we know I'm just going to write it again because again it is an equation you should try and remember so if we're doing the sum of the first 24 months we have 24 over 2 times 2.5 Plus 24 minus 1 and D was increasing by I didn't actually write that one down 0.16 and that again that's why you should write these down all the time because I wasted a bit of time looking back through the question and confusing myself so again keep in mind if you're in an exam if you don't see D when you've written it down you might not even check the answer again and it can cause pain so we have 24 over 2. you can also work this out in your head and then sub it in so 2 times 2.5 I could just put five but yeah it's fine I'm gonna put it in fully just so you guys can see that I'm definitely doing things right it also lets me see like for example if the answer didn't make sense to me I can quickly see where I've gone wrong if I type the whole thing out whereas if I did 2 times 2.5 and I did something weird like right before or something it'd be harder for me to pick up if I don't see the calculation like I do here like I can see 24 minus one yeah that looks correct that's the same here so that is 104.16 tons we're going to keep it as accurate as humanly possible and we're not going to forget the units because examiners are very touchy about that and the sum for the so actually we should write out the ratio as well 1.06 remember six percent increase means you're going to add six percent onto a hundred percent which is 106 turn into a ratio by dividing by 100 or decimal and the sum for this is a R to the power of n minus 1. over R minus one so if we sub those values in a is 2.5 then we have 1.06 to the power of 24. minus one over God 1.06 minus 1. cool so let's type this in we have 2.5 1.06 power of 24 minus one those brackets over you could put this in Brackets but honestly it doesn't matter um but you don't want to work it out 127.04 tons let's call it so normally A Part B or like another part of the question would be which one's better but in this case it just says free scheme find the total so I don't need to bother which is nice uh so that's a five marker again I don't think that's too bad I think that's pretty generous considering we've just used effectively one equation for each one and they gave us all the information so we did two calculations for five marks pretty good okay um for June 2017 paper one so an arithmetic progression has a first term of 32 so let's write that down a fifth term of 22 and a last term minus 28 find the sum of all the terms in the progression so the last term thing is a bit more obscure but um there are a few things that we can do with that so the fifth term so remember we have a plus a minute so I'm actually going to write n equals five let's make it super clear plus five minus one and then whatever D is should equal 22 so subtract 7 so we have four D and then subtract 32 would give me -10 so D is equal to minus 2.5 cool and then the last term of minus 28. so if you remember rightly when we want to work out the sum of all of the terms I need to know how many terms there are so we need to actually work out what the last term thing means you might be wondering also how there can be a last term so keep in mind series can end whenever you want right so the person who made this progression just ended it because they wanted to um so let's just keep in mind it doesn't actually reach an end arithmetic progressions can go on forever right like if I'm adding one each time I can go on forever right you can go on an infinite amount of time now I'm going to write n equals question mark because the last term I don't know what number that is so we have 32 Plus n minus 1 times minus 2.5 equals minus 28. so minus 32 from both sides and expand the brackets we have minus 2.5 n plus 2.5 equals um minus 60. yep cool I don't know why that was so difficult for me n equals minus 62.5 so just use that last bit we have n is equal to 25 it would be so now we can actually do our sum here 25 2. 2 times a which is 32 plus 25 minus one times minus 2.5 again this is where brackets come in handy because you actually remember to multiply by the minus number instead of accidentally adding it or subtracting it in this case which believe me happens a lot more than you might think 25 over 2. multiplied by 2 times 32 Plus 25 minus 1. 2 50. that's kind of anticlimactic wasn't it but yeah it's just a 50. okay and again that's a four mark question I don't think that's particularly bad um but yeah that should be good and then for Part B now we actually have a little bit of a situation so each year a school allocates a sum of money for the library the amount allocated each year increased by 2.5 percent of the amount allocated in the previous year again this is just compound interest so this is definitely geometric and the increase is what 1.025 in 2005 the school allocated two thousand dollars so a is equal to two grand find the total amount allocated in the Years 2005 to 2014 inclusive this is actually a part where a lot of available students make a mistake which is um it's kind of funny but also it makes complete sense finding the difference between years and especially if they don't say inclusive if they don't say inclusive then it's not too bad so if this wasn't inclusive the way to work out how many years that is you do 2014 minus 2005 right and that's nine years right but when it's inclusive you're including 2005 and 2014. so 2005 678 9 10 11 12 13 14 it's 10 years so many times students have actually made this mistake in finding the years and to be honest I don't blame them it can be a bit tricky with the language but remember if you're counting something and including the numbers you need to add one to whatever you normally count right so normally I'd count 2006 seven eight nine ten blah blah blah which should only give me nine so be careful with that they also like to do it with like birthdays so someone might receive a sum of money on their birthday and increases by a percentage um this starts when they're 10 ends when they're 20. how many years has it been well you have to include 10 and 20 so it's not 10 years it's 11 years so n is 11 not 10. which again it's really easy to lose track of if you're ever stuck you know be humble and just literally write out the years as a fight and then count the number of years you have because again with birthdays and stuff people get that wrong all the time um and it's not really necessary so in this case we're actually working out n equals 10 not 9. the other way to think of it is you find the difference in add one right up to you and which one you want to do so again s of 10 we have uh just to remember it's a r to the N minus 1 over minus one so if we write this out again we have two thousand and R is 1.025 to the power of 10 subtract one over 1.025 minus 1. do this two one two three Open Bracket 1.025 to the power of ten minus one close bracket over 1.025 minus one and keep in mind it's money so only go to two decimal places if you go to less than that or more than that they can deduct you a mark and that's my perfect dollar sign we have 22 406 dollars and seventy six cents which brings us to the end and that's all you really need to know for Series in terms of paper one and again if you notice we use the same equations over and over and over again you just need to be able to recognize the equations use the basic kind of nth term and summation equations and then obviously the sum to Infinity assuming it's a convergent series and that's all you really need to know for this so I'll see you in the next one