 So this would be the last session on binomial theorem. In fact, not the whole class Initial 15 to 20 minutes will finish off a very small part of binomial theorem Which is left off and that is basically finding out the binomial expansion when the power is or when the index is any real number Okay, not necessarily a whole number. It could be a fraction. It could be a negative integer So most of you have already done this concept in our bridge course Which we had in the beginning of this year. So I'll just quickly cover that concept and move on to a new chapter today So the name of this topic is binomial Expansion Binomial expansion for any index For any index Okay, so if you have a term which is 1 plus x raised to the power of n where n could be any real number Right need not be whole number. By the way, whatever I'm going to write it is equally valid for whole number as well But it need not be whole number. It could be like say a fraction or a negative integer then this expansion goes like this 1 plus n x plus n n minus 1 by 2 factorial x square and n minus 1 n minus 2 by 3 factorial x cube and so on dot dot dot Now dot dot dot doesn't mean we'll go to infinity, right? It means it will go to the extent which n allows it to go Okay, if n is a whole number If n is a whole number, then this would be going to enter n plus one terms, right? We already know that So let me write down over here So it would have it would be a finite. Let me write it down over here. It would be a finite series Series if n belongs to whole number So we all know that when n is a whole number or when the power is a whole number The total number of terms in a binomial expansion like this is supposed to be n plus 1 So that would be a finite binomial series, right? They will have limited number of terms But if n is not a whole number that means if it is a integer Which is negative or it's a fraction Then the number of terms would be infinite. It would be an infinite series Okay, let me give you an example for this. Let us say I have 1 plus x to the power of Let's say minus 2 Okay, so when I use this particular series for it, it will be 1 plus n is minus 2 x then n n minus 1 by 2 factorial x squared then n n minus 1 n minus 2 by 3 factorial x cube now here you would realize that These terms n n minus 1 n minus 2 will never reach to an extent where it'll start giving you zero In fact, you can see this negativity is increasing from minus 2 minus 3 minus 4 minus 5 minus 6 It's always on the rise That means it is not becoming zero ever In case of whole number There used to be one stage where you would realize that it will become a zero for example had it been just two then two two minus two Two two minus one two minus two that would become a zero That's why our series used to stop in those cases where our power used to be whole numbers But in such cases our series is never going to stop Okay, it's going to be go on and on and on right so you may expand it Properly and write it but my my aim is not to give you the expansions of this My aim is to just tell you the structure of this expansion Okay, so try to understand the difference here the difference is This is a universally applicable binomial expansion If n is a whole number it would be limited to n plus one terms That means it will stop somewhere that means it will be a finite series But if in n is not a whole number means it is a negative integer or let's say if it is a fraction Then this series will go on and on forever. Okay, so there'll be infinitely many terms in that One important thing that we need to keep in our mind is that Whenever we are writing such a series where the number of terms is infinite Remember mod of x here should be less than this number one Okay, only then our series will be convergent in nature Okay, and then only such expansions make sense I'll repeat once again If your number of terms on the right-hand side is in is infinitely many and your series is not convergent Convergent means even if there are infinitely many terms the sum should be finite So if that condition is not fulfilled, then this expansion will not hold good Are you getting my point? Okay, so what will happen right-hand side will become infinitely big in terms of magnitude Left-hand side would be a finite term This is something which I had already discussed with you during the bridge course when we were doing limits standard algebraic limits Right, so this condition is very very important for the convergence to hold true Okay, how many people ask me said you have written for one plus x What if there was a plus x doesn't matter if you have a plus x if you have a plus x To the power of let's say n All you can do is you pull your a to the power n outside and write it like this Okay Now why does I advisable to make it in this form is because when one of the terms is one It's very easy to write down the expansion Right because you don't have to care about the power that you are going to put on one that anyways is going to be a one Right, so here we'll write it like this one plus n x by a Okay, then n n minus one by two factorial x by a whole square and So on and so forth. Okay, again It will be in finitely many terms if your n happens to be a negative integer or a fraction right, in fact, it could be irrational numbers also and It would be a finite number terms if n is a full number, right? And here the condition that would be required for convergence is Mod of x by a that is this term should be less than one You may also write it like this mod x is less than mod a Only under this situation you will be able to make this series of convergent and hence a valid one Okay Now this concept is not very very commonly tested concept But yes, it's good to know it you never know a question may be framed on it. Okay, so the idea is Just use NCR Expansion mode to write this term. That's it. There's no other concept involved and yes Make sure you remember the condition for convergence. That's it in in terms of the combination Combination is used in binomial expansion. You don't have to justify binomial expansion in combination NCR doesn't make sense if NNR are not whole numbers Okay, so binomial Uses permutation combination idea not vice versa. Don't get it wrong It doesn't mean that these terms have to come from NCR only no Okay, NCR was used here when your n was whole number, right? Because those terms were already introduced Before we started this chapter, but vice versa don't think like that that okay I have to have this as some NCR. This is NC, you know NC1 and NC2 etc. No NCR makes sense only when NNR are whole numbers. Okay, I'll see do not make sense Now what kind of questions can I get on this? Let us try to get our hands on some questions Not take much time. I think two or three questions is good enough for us Let's check if I have some questions based on this binomial theorem Okay, let's start with this question if this term is approximately equal to a plus bx Okay For small values of x. Okay small values of x because they want to meet that convergent situation Then what is the value of A and B? Let's try this out. Everybody please work this out. I am putting the pole on I think two minutes is good enough for time Meanwhile, let me check the attendance. I think people have joined in Let me take the attendance Okay, one more thing before I forget Has your school declared holiday for 14, 15, 16, 17? Is there any school or is there any people from school over here whose school has not declared holiday? Why it's a season of Pongal and What do you call this? I forgot the name By Sakhi and all we call it. Yeah no NPS people I don't think so SSRVM has their own video Okay, I think Pongal falls Pongal or this thing Yeah, 14th. Yes, 14th. Never mind. Okay. Keep me updated. Keep me posted five four three two one Go, please vote. Please vote. Very good. One of the options have by far Exceeded the others and that is option number B. Let us check Alright, so let's take each one of these terms. Let's say the first term Let's say the first term One minus 3x to the power of half, right now since you have to approximate it till The Constant and the term containing x we can write it as 1 minus half Or you can say 1 plus half into minus 3x you can write it as 1 minus 3x by 2 I'm sure you would have used these kind of approximations in your gravitation chapter for finding the value of G for below the earth, right? If you go H Meters below the earth, what is the G due to gravity? Yeah Yeah, so let's write down the next one 1 minus x to the power of 5 by 3. This will become 1 minus 5x by 3 Okay, so on the numerator you will end up getting 1 minus 3x by 2 plus 1 minus 3x by sorry 5x by 3 5x by 3 correct in The denominator before you start using anything take 4 out 4 will come out as a 2 and You will have 1 minus x by 4 to the power of half Okay, so let's see how we can simplify this. This is nothing but 2 minus Minus 3 by 2x and 5 by 3x That's going to become minus 19 by 6. Isn't it? 3s minus 19x by 3 Okay, this 2 I will write it as a half and this I will write it as 1 minus x by 4 to the power of minus half Okay, so this will become half 2 minus 19x by 3 and This will become 1 plus half x by 4 Okay So half let us try to see this will become x by 8 Let's write this as x by 8 Okay, so 2 into 2 will be 2 2 into x by 4 will be sorry 2 into x by 8 will be x by 4 minus 19x by 3 and Other term you don't have to write because anyways you don't require x square term You only have to write out terms till x correct. Yeah, so this will be half 2 This will be nothing but 3 minus 76 which is minus 72 by 12 minus 72 by 12. So minus 70. Sorry Yeah, have I missed out anything? Oh, this is 19 by 6. Oh, sorry. Yeah, so this will become minus 70 x by 24, okay on opening this term you will end up getting 1 minus 35 By 24 x Okay, so a is 1 and b is minus 35 by 24 which option matches with this which option matches with this I think option number b is the right option. Well done. Well done. Janta first question Good Is this clear how it works? One common mistake I have seen people doing When you write your binomial expansions, we normally make this term as one Let me write it. Yeah, this term is one. So this term should be brought out of that power When you bring it out Please note it will come out under the effect of that power. So 4 will come out under the effect of power half So that will come out as a 2 Right, so I've seen that mistake happening a lot with students. They just bring out the four term out Forgetting the fact that that four will come out under the influence of that power Okay Can we move on to the next question anybody who wants to copy this? Please do so so that we can proceed done Okay So let's take another one the value of This expression is which of the following Let's put the pole on Let's put the pole on Three minutes. I think three minutes is good enough for time for this question So it's an infinite series which goes like one plus one-third square plus some weird terms are coming up Indirectly the question setter is asking us whose binomial expansion is this still very good I've got one response from somebody last 40 seconds Okay, five Four Three two one go Take a guess take a guess even if you're not getting the answer no problem Just try to see whether your guess work is you know working or not, right? I always keep on saying Think as if you're answering a CT This thing CT pattern question paper remember CT there is no negative marks So those who will do those who are planning to write KCT exams There's no negative marks in CT exams. They should mark all the questions and come right never know of course those Which you have solved will definitely be correct those you haven't been able to solve that also should take a guess Okay, now here most of you have said option number a Option number a let's check whether a was correct See if you look at this pattern This is a question which is like to be solved by observation you have to observe a lot, right? These terms are definitely factorials two factorial three factorial and there is always the presence of one-third one-third So one-third is basically increasing so something related to One of the terms being one-third. Okay. Now. Let's analyze this into more detail This is nothing but one-third of one-third. That's how one-third square pearly would have come probably would have come I'm not taking a guess And I have one four seven Now one four seven will only come when there is a One-third four-third seven-third coming up because you have to subtract the one one each Isn't it? So if I write this like this one third four third two factorial one by three square And remember if you're subtracting it cannot become four-third unless until this is minus one-third Are you getting my point? see the analysis So what I did was I Realize one four seven etc is coming right isn't it? So from one by something To reach four by something Right this guy and you have to subtract one from it Correct. So this guy has to be a three kind of a thing Yes, I know then only we'll get from minus one by something minus one to minus four by something Okay, so that term has to be a three Are you getting my point? So here also have to go and make a change. This is minus one-third. This is minus one-third Right and this is minus one-third square. Are you getting my logic over here? This is very important Okay, so this term that is your third term of the expression you have to restructure it like this Okay, so automatically your fourth term will also fall in place. So your fourth term would be minus one-third minus four-third Minus seven-third which is actually minus four-third is what minus one-third minus one So if this is like n this is n minus one. This is n minus two Correct and one two three is three factorial Right and since you have provided three from this base, you will be only left with one by third whole cube and One by third whole cube Please note. We have to write it as minus one by third whole cube because your net net power is positive over it Your net net power is positive isn't it? So if you don't write one minus one by three over here your net power will become negative Yes or no? So from this particular breakup we can figure out that oh this particular Expansion or this particular series was the expansion of one minus one by three to the power of minus one by three In other words, you were expanding two by three to the power of minus one by three Which is nothing but three by two to the power of one by three Is this there in the options? Yes, option number a option number is correct. Janta is absolutely right Okay, anybody who got this right by sheer guessing Very good Actually doesn't make a difference half guessing for Aditya. Okay. Yeah Does anybody want me to explain this again Aditya? I just saw your message You do you understand the logic now? Shares Shares also if you expand, okay So Harry Haran is a question one plus half to the power of one by third. You're saying that How can it this be because the second term would have been one third into half? No, I Have a one-third cube. Oh, sorry. I have one by three square This some one match See you are not comparing term by term the cumulative expansion will be the same Are you getting my point if somebody says From this series figure out whose expansion was it? I would say this result Of course, this result will also give you the same but the cumulative expansion not the individual terms Don't compare the individual terms If you add the whole thing that result would be the same as when you expand this game got it? Yes, yes, I'll explain once again. She says again. Look at the pattern here The pattern has one four seven and you have to your factorial of number and you have one third repeated You know as powers of one third, right? so as a person If I have to look at see this term is what this term is like nx Okay, this term is what n n minus one by two factorial x square Okay, this term is what n n minus one n minus two n minus now see it From this term Looking at one-third presence. You can make a guess that it could have been One-third of one-third or it could have been minus one-third or on my into minus one-third, right? Yes or no the presence of one four seven see when does a fraction having a numerator of one on Subtraction of one will give you a fraction with a numerator of four Correct. See this to this When will let's say n is a fraction you're subtracting one from the fraction and you're getting a four on the top Right, so one four seven will come only when there was a minus one by three and you subtracted a one from here You would get a minus four by three Again if you subtract a No, two from here. You'll get minus seven by three. So the presence of one four seven comes because You had n as minus one-third. So this is ruled out and this is accepted So this first term Sorry, the second term was minus one-third into minus one-third Third term. This is two factorial. This term is your two factorial. So this term compare it with this term and One four just provide it with a three three from this guy so I Just write it down. So one four One two one by three to the power four. What did I do? I wrote it as one-third four-third one-third square by two factorial and I placed minus sign over here because without a minus We cannot get a minus four by three on subtraction of one Are you getting my point? This is the part which we need to understand See looking at this. This is the only possible answer Right This can have multiple expansions This can have multiple expansions, but this answer would have only one result They cannot do multiple results Okay Alright, let's look into one more question and then we'll close this concept coefficient of x to the power two Power m plus one x to the power two two is also raised to the power of m plus one in this expansion Is which of the following? Mod X is less than one Let's have around two and a half minutes two and a half minutes Time starts now very good last one minute last one minute 30 seconds Okay, five four three Two one go Okay, Pranav, what is the option that you think is correct? Let's hear it out from Pranav Okay, Pranav thinks B While Janta here thinks A. Okay, let's see whether Janta is correct or Pranav is correct All right, so first of all I would like you to Meet this very Special term this you will keep on seeing in very Very commonly in other chapters also See what is happening here constantly your powers of X are becoming raised to Higher powers, right? So in such kind of questions, what do we do is we initiate a chain reaction by multiplying and dividing with one minus x Okay, I'm sure most of you would have started applying binomial expansions on each one of these terms by taking it to the numerator but before that if you could do this your life will become super easy and this would become a Hardly 20 second problem, right? If you do that you would realize in the denominator this guy will become 1 minus x squared This guy will become 1 minus x to the power 4 This guy will become 1 minus x to the power 8 This guy will become 1 minus x to the power 16, right? And da-da-da-da till the last guy will multiply and give you 1 plus x to the power 2 to the power m plus 1 Okay, so this whole term will boil down This whole term will boil down because of this chain reaction to 1 plus x by 1 plus x to the power 2 to the power m plus 1 Which we can write it as 1 plus sorry 1 minus x into 1 plus x to the power 2 to the power m plus 1 raised to the power of minus 1, right? And this is quite simple. We can apply our You know by normal expansion on this Yes, hello, so it'll become 1 minus x and 1 minus x to the power 2 to the power m plus 1 and other terms would be higher Other terms would be of higher power. Okay, sorry for the sign change. This is minus by mistake Yeah, so this will become plus okay now the coefficient where The coefficient of x to the power 2 to the power m plus 1 is what I need read the question question They want us to find out coefficient of x to the power 2 to the power m plus 1 So that will be nothing what this multiplied with this because any other term would give you Powers of x more than this So the answer would be the coefficient would be 1 in this case that is when this guy Multiplies to 1 over here. Okay. No other coefficient will appear other than this one. So Pranav is Absolutely clear. Oh, sorry. Pranav is absolutely Correct. Why did I say clear? Is it fine? Any questions? Okay, Janta was wrong Pranav was correct Okay so with this we you know complete the Binomial expansion chapter now we are going to hide towards a new chapter