Tutor Belloᵈʸᵈˣ is live

Added: 2026-05-20

[00:00:02]I just I cut it down just Thank you.

[00:01:04]Can we all see the board? Can we see the board?

[00:01:13]Can we see the board?

[00:01:22]He's on the side.

[00:01:52]Just passing out.

[00:02:20]I should be able to = 1 + the right side isn't it?

[00:02:53]So like I said tan a - tan / 1 + tan.

[00:03:19]So we assa tals tan / 4us what tan / I'm not dealing with the right hand side.

[00:03:59]So tan / 4 just like tan a - tan tan isn't it? And we know that 45 1. So this guy give us 1us tan theta / 2 / 1 + tan tan is what susan theta / 2 is going to give 1 sin theta / 2 / cos theta / 2 cosine / 1 + sin theta / 2 / let's square Let's actually 4 - 2 is going to give us what the square of this.

[00:05:53]So you know if you have a b a square going a square so we going to have 1 square us 2 * 1 * right going to give us 2 * 1 2 sin theta / 2 cos what theta / + b² going to get what? Sin square.

[00:06:40]I said by 95.

[00:07:13]Can we continue? So here we are just now right 2 / what cosid What?

[00:07:38]2 / two. Is it clear that part?

[00:07:48]Now we are checking at the right hand side. The right hand side determine what I'm going to do. So my rightapide I cannot be dividing by going this guy will go this one will go this will go this will go something like sin everywhere. So I'm good to go. Now let's I mean multiplication rather not division multiply the numerator and just multiply everything by one and I can write one to be what cos² theta / 2 / 2 isn't this one so I don't change but I want to write my cos² I have what cos Let me just denet square the 2 * this this will cancel one out of it, right? So we have 2.

[00:09:33]This time this will cancel this. What are we going to have sin square?

[00:09:43]Same thing we have here. Just 2 sin theta / 2 cos theta / 2 + sin theta / 2. Is it clear that any question?

[00:10:02]Let's check very well. Ask your question. Is it clear?

[00:10:10]Which part?

[00:10:29]Yeah. This time you have multip inside 2 cos theta / 2 sin theta / cos theta* cos² theta / So this any question?

[00:11:16]>> Okay. Okay. I'm sorry. I'll try not to write that part. Let me just No, for the sake of let me see the principle. How frustrating.

[00:11:39]My eyes.

[00:11:44]I think I have a better writing.

[00:11:48]Don't worry. Everything starts sin.

[00:12:23]Now for the right hand side for the right hand side.

[00:12:42]Now note tan a minus b = a - tan b / 1 + tan a / 1 + tan a tan.

[00:13:03]So that means tan<unk> / 4 - 2 should give us what tan 4 - tan theta / 2 / what tan 4 tan theta.

[00:13:27]Now let's square that.

[00:13:30]Now square both sides.

[00:13:34]Square both sides.

[00:13:37]So this is going to give us tan square<unk> / 4 - 2als.

[00:13:48]So this is already one this one tan theta / 2 / 1 + 1 tan theta / 2 / 1 tan 2. Now we square this square. Now what are we going to get?

[00:14:28]So this is going to give us a square that's right 2 * 1* we have 2 / 2 right square B square we have tan square theta / 2 isn't it divided by 1 + 2 * 2 + what?

[00:15:08]Is it clear that part?

[00:15:13]>> So now um note note tan= s what?

[00:15:24]So we are going to change all this terine to s and cosine function of s and cosine. So these guys give us what 1us 2 sin theta / 2 / what cos theta / 2 plus what?

[00:15:41]Sin² theta / 2 / what cos² theta / 2 sin² theta / 2 / cos² theta / 2 everything 1 + 2 sin theta / 2 / cos theta / 2 + sin square theta / 2 / cos² theta / 2. Do you get do we get to that part?

[00:16:31]Do we get to that part?

[00:16:36]Okay. Okay.

[00:16:38]It's already procid by cos 2. So we use all the numerator to multiply this 1 * square that's what this* remove this We only have one of these here. 2 * 2in cos theta.

[00:17:41]This is square divided by sameator.

[00:17:59]Plus what 2 sin theta cos theta / 2 plus what sin square what theta is it clear that part >> to make it bigger right I don't give us I is it clear to this part so now let's combine join um these two together.

[00:18:30]These two together. So here we have what? cos theta / 2 cos square cos² theta / 2 + what sin theta / 2 - 2 sin theta / 2 cos theta / 2 / cos² theta / 2 + sin² theta / 2 + what 2 sin theta / 2 cos theta / 2 is it clear I join this this first and last together we know trick identity this guy to one isn't it it's not necessarily it must be just that the two angle must be the same so like sin square + cos square is 1 sin square x + cos x + cos 1x 1 is So since this is like the identity so here we have 1us now before then note okay let me write it out note cos² theta / 2 + sin² theta / 2 = 1 and 2 sin theta / 2 cos theta / 2 = what sin theta I hope you know this okay the one we are familiar with is 2 sin theta I mean sorry. Yeah. Sin 2 theta equals to what? 2 sin theta cos theta. We remember this. But if you if you if you divide the whole angle by two / 2 / 2 / 2 this one 2 we have sin theta left.

[00:20:20]Right. Right.

[00:20:22]>> So we have sin theta= 2 sin theta / 2 cos theta / 2. That's why we have this 2 sin theta / 2 cos theta / 2 cosin theta.

[00:20:31]Do we get what I did?

[00:20:34]Do you get what I did?

[00:20:36]You know we are familiar with sin 2 theta= to 2 sin theta cos theta. Now all the theta divided by 2. This sin 2 theta will now remain sin theta which is this.

[00:20:47]All this 2 sin theta cos theta will become 2 sin theta / 2 cos theta / 2. So sin theta is equal to 2 sin theta / 2 cos the 2. If you want to turn it back to this multiply the angle by 2. We have sin 2 theta= 2 sin theta. So that's that for that. So now at the end of the day since this guy we have 1us what sin theta what 1 + what sin theta. So that's the that's the food. Is it clear?

[00:21:24]>> Check from beginning everywhere we multiply. See we multiply everything here by cos square the power 2. So cos square * 1 we have cos square right cos square theta / 2 * this this like cos² theta / 2* 2 sin theta / 2 cos theta / 2 2 sin theta / 2 / cos theta / 2 like if you multiply this by this this I've written it out no this one we cancel one out of this you Now cos theta 2 * 2 sin the power 2 now give me 2 sin the power 2 cos theta / 2.

[00:22:16]Now check this again cos square * sin square the power 2 / cos square.

[00:22:23]This one is square and this is square.

[00:22:26]All this we cancel this. You have sin square left.

[00:22:30]Same thing the one down use to multiply all this one again just like the one we did. You also get this.

[00:22:38]So here now just pick these two together.

[00:22:42]Let this one stand alone. Pick this first and the last together. Let this one alone. Since this guy is one then this equivalent to sin theta. This guy is one. This also equivalent to sin theta. Is it clear?

[00:22:58]Let's let's write it before I clean.

[00:23:00]Let's write. How are you?

[00:23:16]>> Okay.

[00:23:24]This thing is is go to WhatsApp now. Go and tell others to join us cuz I cannot call the video again to go and >> where where this note this let You know sin 2 theta equals to what?

[00:24:08]2 sin theta cos theta right. Now divide all the angle by 2. Divide all the angle by two / 2 / 2 / 2. What will this one become?

[00:24:24]Sin theta right equals to what? 2 sin theta / 2 cos theta / 2 isn't it? So since this is 2 sin theta / 2 cos theta / 2 you can replace it with sin theta. So that's 1 - sin theta / 1 + sin theta.

[00:24:47]So deduction is very simple but I don't like some of those stress but let's just do it. By the way now let's write it down before I clean the body. Let's write it down.

[00:25:01]We shouldn't spend more than an hour on 101.

[00:25:06]So after we do after the assignment on we do 102 assignment then we now come back to like normal class.

[00:25:15]Can I play the board?

[00:25:18]Let me see. Ah, nothing I don't have in this life is writing. I don't like what I'm saying.

[00:25:26]This light is not finally.

[00:25:34]Ah, God have mercy. Don't worry. I'll be I'll be better one day.

[00:25:40]Can I clear the board?

[00:25:43]Yeah. 2 minutes. 2 minutes. Let's be fast.

[00:25:53]I hope we can all see it. All those that are streaming online, can we all see the board?

[00:26:15]Most of them they are slept though. They are streaming on your bed as >> we can't see the body.

[00:26:36]>> Maybe I just What do you say?

[00:26:41]>> Maybe you just write in the video.

[00:26:43]>> I'm coming. Let me see if maybe if bending it can work.

[00:26:58]Look.

[00:27:07]Okay.

[00:27:25]No.

[00:27:40]I'm waiting for us to write everything so that I can clean the body.

[00:27:50]See phone auto rotate.

[00:27:54]Check auto rotate. Let me see how it look like.

[00:28:16]Okay, they said it's better now. Fine.

[00:28:18]Thank God. All right, you can follow up then.

[00:28:22]Can I clean it now?

[00:28:25]Can I clean it?

[00:28:27]All right.

[00:28:29]Okay.

[00:28:44]I hope you have all written it.

[00:28:48]Some class rep. Who is class rep?

[00:28:52]You're part of those ones who submit tomorrow.

[00:28:59]shake during my level where there's no assignment I've submitted like before the due date I mean I will come down everybody write my own like I used to write assignment class they all know me I will teach everybody they write then I now turn my own so sometimes We submit after the dates since I'm the rep they will collect it.

[00:29:30]I've submitted assigned like 3 days to arate and two people two people were left behind. He told me that I should go and collect it but they should give me money and all those kind of used to crack actually but can be very frank at time.

[00:29:46]Now what's the second question?

[00:29:50]Show that.

[00:29:55]Show that what? Summation as K ts from 1.

[00:30:01]As K from 1 to N of K^ 4 equals to what?

[00:30:07]N into what?

[00:30:09]n + 1 2 n + 1 3 n² + 3 n + >> - one I try at least for over what that's the question is it now you can solve it directly and but I prefer that you do it like you first interpret this guy so that it can look so familiar That was what we've known so far. So first interpret this guy. The meaning of this guy is that you are doing summation of discrete number like k = 1 2. Whenever you see a sign like this is counting number natural number 1 2 3 4 5 whenever you see the sign. So unlike this sign this one now you are counting all real number like depending on the intera given to you. If I put one here and I put two here all the real number between two and one like 1.01 1.0 like continuous addition. This one is discrete addition like 1 2 3 4 5 6 depending on where you are starting from. So now our k starting from one is going to end. Now if k is one what will be this one right? Since this means summation then you have if k gets to two what will be this 2^ 4 right when k gets to three what will be this but there's no time to write everything just write dot are we together when k gets to n what will be this n^ 4 equals to what n into n + 1 2 n + 1 3 n² + 3 n - one over what?

[00:31:55]So this is the question right this is the question either this or this you can solve it though you can do this directly but it's better you first interpret this aspect of this now like I said on the left hand side what you are doing is summation on the right hand side what you doing is substitution for instance what do I mean by that you say your n is four on this side you doing summation of the first four on this you are doing substitution of four just substitute four to wherever you see n on this side you doing sum of the first four terms. If your n is two on the left side you doing summation of the first two terms on this side you are substituting two for n. So on the left hand side you are doing summation of 10 on the right hand side you are doing substitution of n. So now for your base case these are assignment not j. So all these notes you must write it out. Base case.

[00:32:48]Best base case show that it is true for what? N= to 1. So this should be the condition n as element of natural number. Those they did not write it but that should be the condition. Most cases this is the condition given. So the base case that show that it is true for n= to one. Now if n= 1 that on this side that the summation of the first term alone right which is 1 = let's put n here 1 into 1 + 1 2 2 * 1 >> + 1 >> 3 * 1 square that's 3 + 3 - 1 >> 5 right >> over what?

[00:33:36]So here we have 1 = 2 * 3 * 5 >> 30 >> 30 cancel 30 right so 1 = 1 we conclude the statement the statement is true for what n = 1. So we are done with the first case. Now the next step is our induction hypothesis.

[00:34:02]You write it out induction induction hypothesis.

[00:34:10]So what do we mean by hypothesis is like an assumption right? So we assume assume is true for a particular big number big number k to be precise.

[00:34:21]Assume it is true for n= to k. If you have your time you can test for n is 2. M is you love stress but once you've done one and it's true just move to direct I just step out okay base induction hypothesis then your induction step then you are done now assume it is true for n= to k we assume that k is a big number so in this you summation of the first k which is going to include all the big number you assuming is including all the am I communicating m without. Now assume is true for n= to k. Now here we are doing summation of the first k terms which is 1 + 2^ 4 + 3^ 4 plus dot dot dot plus what n^ 4 but you put k there k^ 4 equals to what?

[00:35:21]n which is k into what? K + 1 into what?

[00:35:27]>> 2 k + 1 into what?

[00:35:30]K square + >> - 1 over what?

[00:35:36]>> 30.

[00:35:39]Now you cannot move beyond this for there's nothing you want to solve again because there are missing things here.

[00:35:44]Now because you written this expression out then actually true for any case your assumption is true.

[00:35:52]Any assumption in math actually true.

[00:35:54]Assume it is true for y= k which we've shown. Then you conclude that if true now your induction step your induction step now your induction step we now include if true if true for n= to k. You know initially you were assume but now you've concluded that if true for n= to k then show that it is true for what n= to k + 1 once you able to show it then the statement is true for all natural numbers of n are we together so let's write this down so that can I give us 2 minutes to write Everything just stop there because I'm going to clean everything all at once.

[00:36:48]I hope it's clear to that part.

[00:36:50]>> Are we sure?

[00:36:56]>> Can we all see the board? I hope we can also see it too.

[00:37:09]>> Yeah, I think it's better.

[00:37:11]It's kind of better.

[00:37:14]I always say we don't sleep.

[00:37:19]So now don't sleep.

[00:37:39]Okay. Okay.

[00:37:49]Okay. I give us 2 minutes to write that before I clean it. That's number two of the 101 assignment. 2 minutes to write it. Or can just screenshot your phone since you are watching. You can screenshot and write it later.

[00:38:07]Anyone like, "Have you have you any mosquito bite.

[00:38:45]If if you are now like 500 mosquito, you have seen their blood.

[00:38:52]Am I lying?

[00:38:55]All 10 students here now find solutions to your lecture theater.

[00:39:01]The place where I manage they lock it for days.

[00:39:05]But man, imagine.

[00:39:11]Can I have one more minute?

[00:39:20]We are not going to waste time after this. We have one left. One question left.

[00:39:27]>> The stress in this question is just the expansion of >> Please don't sleep cuz one is the Who is the only one here?

[00:39:46]>> Raise your hand again. Only one.

[00:39:57]>> So when we are done with 10, we actually come back to 101 to treat some topics.

[00:40:03]But we don't know the assignment.

[00:40:12]I just started our our exam. We start our exam next month.

[00:40:22]>> My exam TV will start next month.

[00:40:27]>> Wonderful TV. 10:00 p.m. to 5:00 p.m.

[00:40:31]every day.

[00:40:37]Is that >> Is that bad?

[00:40:40]>> Can I hear it now?

[00:40:42]>> I can hear it.

[00:40:57]>> Sometimes sometimes we be here we have exam tomorrow.

[00:41:02]I have like most of us would have you know have done what whatever I want to do before we come for TV and as we are ending the TD we just walk straight to TLC last year last year last year we found even physics 102 >> from that other room to even m1 from that room to math department.

[00:41:33]Now can we we stop at where? No, let me write the K term first. 1 + 2^ 4 + 3^ 4 + dot dot dot + K^ 4 = K into what? K + 1 2 K + 1 3 K² + 3 K -1 over what?

[00:42:01]Over 30. Now we concluded earlier that if true for N= K then show that it's true for N is K + 1. Now assuming K is 10 just assuming K is not actually 10. Assuming K is let me say assuming K is 9. K + what?

[00:42:18]10 right then you know summation of the first four term first 10 terms rather is going to be what summation of the first n plus the 10th term isn't it and I say a b c d summation of this whole term is going to give us summation of the first k plus what the fourth term so the summation now I want to do summation of the first k + 1 step because now we are saying n= k + 1 right so summation of the first k + 1 terms don't write this again we've written it already summation of the first k + 1 terms which is going to first of all include summation of the first k terms this what I want to write next now is where you should start when you writing this going to be 1 + 2^ 4 + 3^ 4 plus what do plus what k^ 4 of the first K term plus the K + 1 term all together is going to give a summation of the first K + 1 term. Do we get like like I said earlier A B C D this fourth term summation of this fourth term I mean this first four term or let me add one more EF now there's a summation of the first four term it say now summation of the first three term plus the fourth term right >> do we get that summation of the first fourth term first fourth term equals summation of the first three term plus the fourth + 10. So summation of the first k + 1 term is going to give us summation of the first k term plus the k + 1 term. Is that clear?

[00:44:04]Like we want to sum all the term from the first term to k + 1 term. Since k + one is the next term after k going to give us the summation of the first k + one term is going to give us summation of the first k term plus the k + one term. Now what's our k + one step? I write that in form of n the last term in the question which is what n^ 4 isn't it?

[00:44:28]>> But now what's our n k + 1 by what?

[00:44:32]>> If this is the k term this will be our k + one term. All this one is our first k term everything first k + 1 term of which our first k + 1 term equals to first k term plus what k + 1 term. Are we together? equals to k + one. So from the question we have n here since our n is k + 1 then n + 1 k + 1 + 1 that's what >> k + 2 2 n + 1 2 n + 1 2 * k 2 k right 2 * 1 2 + 1 3 then we have 3 n² + 3 n - 1. So here we have this guy is going to give us 3 into what? k² + 2 k + 1 + what? 3k + what? 3 * 1 3 - 1 2 3 * k square >> 3k² 3 * 2k don't need to show all this.

[00:45:43]3 * k² 3k square 3 * 6 2k 6 >> 3 * 1 + 3k + 2 Yes. So here we have 3k² 6k + 3k >> 3 + 2.

[00:46:02]So you can just write it direct don't need to show the I'm just writing out 3 n square + 3 - 1 if you expand it you get this.

[00:46:10]So that's that for that over 30, isn't it? Divided by what?

[00:46:18]>> 30.

[00:46:20]But we can't solve this guy because we cannot write all the terms out here.

[00:46:24]There are many things natural number.

[00:46:27]But even though you are going to count from 1 to 1,000, those one are even much enough. Now, but because we have an identity for this guy, you know, this guy is equal to this, isn't it? And this is equals to this. Since this one is has finished structure like this guy. So we can replace this guy with this. Isn't it? So instead of writing this, we can write this because this is same as this and this is equal to this. So here we have K into what? K + 1 into what? 2K + 1 into what? 3 K² + what? 3K - 1 over what? Some people used to forget that there's something left here. Please don't forget. + what?

[00:47:08]>> K + 1 by what? 4 = to what? K + 1.

[00:47:15]>> K + 2.

[00:47:17]>> 2K + 3 K² + what?

[00:47:22]>> 9K + what?

[00:47:24]>> Over what?

[00:47:26]>> 30 + >> Oh, sorry. 30. Is that clear?

[00:47:37]>> So now let's find the LCM here so that the division can be under everything. So let's leave that guy. So this guy become K into what? K + 1 2 K + 1 3 K² + 3 K - 1 plus you know what?

[00:47:59]>> 30 like 30 1* this you have this right?

[00:48:03]this over 1 30* you have >> 13 what 4 equals to this guy are we together >> so now okay let's let's factor because we have k let's bring it out I'm focusing on the left hand side now and let still be sleeping now what do we have on both what is to both >> so bring it Left we have k + 1.

[00:48:34]What will be left here?

[00:48:36]>> K. All right.

[00:48:38]>> 2k + one. Right.

[00:48:41]>> K.

[00:48:42]>> 3k 2 + 3k + 1. Is that clear?

[00:48:47]>> If you use this to multiply this back, you get all this.

[00:48:52]Okay. Sorry.

[00:48:55]>> Is it clear that part?

[00:48:57]>> Plus what?

[00:48:58]>> 30.

[00:48:59]>> 30 into what?

[00:49:01]K + 1 cube you know we already factor one out we already bring this one out. So if you multiply this by this by^ 4 so that we use big bracket now showing that this is multiplying everything divided by what >> 30 now we have to open the bracket of everything that is inside.

[00:49:26]So let's just write it. I'll clean the board one more time to show that. So now let's write it before I clean the board.

[00:49:33]Okay, let's solve one more. I'm not writing this yet. I want to put this thing here. So here we have k + 1.

[00:49:43]Now k * 2 k >> 2 k² k * 1 + 2 >> + k open bracket 3k² + 3k - 1 3k 2 + 3k - 1 then we have what + 30 into what k + 1 cube over over 30. Is it clear to that part?

[00:50:20]Eh, I just use this K to multiply 2K + 1 to give me this. Then I I wrote everything back. So now let's write it before I should be included.

[00:50:34]Please write very well at least the 10 is like a bonus.

[00:50:42]Most time this one is greeting good morning.

[00:50:53]It's kind of clear clear.

[00:51:20]I hope you have not slept.

[00:51:35]I have one.

[00:51:37]I use it to multiply 2k now become 2k square k * 2k 2k * no it's k What is this?

[00:52:08]Active sharp.

[00:52:25]Some people have slept. Go to WhatsApp.

[00:52:27]Go and dialogue.

[00:52:30]>> Yeah.

[00:52:31]>> Let me to open the door, please. How >> go no no no no no she's an admin go to go and send message to your friend to join cuz we will soon be done with and I want everybody to be part of the complex number I want to solve next go to drop a message that everyone should join you they don't even need link they should just come on my channel come up to my come on my channel on YouTube and just check for the Live live stream.

[00:53:17]Are we done?

[00:53:19]>> Can I Prepare yourself. We are not starting know when last I've taught like this that was my 100 level. This night I used to be m okay.

[00:54:05]>> Yeah. Yeah.

[00:54:23]Are we done?

[00:54:31]We buy she come and pay all and share for everybody.

[00:54:39]Some people don't want to go. We beg them please come.

[00:54:45]It was funny actually. But then we were we were colleagues like there was no senior no junior like you now you are like junior those days we abuse our there was no t no student abuse our abuse can I can right everything we are good to Shout blah blah blah she now do you know why induction is like do you know why that you can prove k + one do you know why is true for the condition given to you it's because Now the K you assume is like a big number that that like covers everything you've written already. You know when we substitute K it covers all the left hand side isn't it?

[00:55:56]>> Yeah. Now imagine k + one like it's just like he's saying he's saying uh I cannot put it now there is a course we are doing 200 level um physics like um what Isaac Newton proposed like all these classical Newtonian physics if they are only like he said time is time is I'm coming time is um invariance like time is independent time is absolute rather what do you mean by time is absolute I just want to explain something here what do you mean by time is absolute because is that as you sitting down here as I am even though if I'm running like the same time you'll be counting the same thing I will be counting are we together like if you are sitting down even though we on different frame maybe I'm in a train or a train rather or in a car or even in an airplane and you are sitting down by count 10 seconds in an airplane you also count 10 seconds like time is the same everywhere that's classical physics but in in special relativity it's not true for speed of light like true if the speed is far far from speed of light but as you approaching the speed of light time is not absolute for instance um if somebody is traveling with the speed of light it can count 3 seconds and you are still counting 2 seconds. Do you get what I'm saying? So and the Newtonian physics could not prove it for speed of light. He could only use that for our everyday speed like the normal speed car and everything the theory could not be proved for speed of light.

[00:57:43]So when abat testing came that was when the man started the special relativity the special theory of relativity and when testing could could prove all those stuff that we are talking about for speed of light then he concluded that if true for speed of light then is true for everything I know for the classical phys greater than speed of light but as v is approaching speed of light all the classical physics they not true again like they are false everything that they are false as velocities approaching speed of light so but that's when when this guy say um t= to t prime that this is why in another frame and I'm in another frame we me the same time but this guy said t is not equals to t prime said t is not equal to t prime and he was able to prove it to and he said because um how did he put it the speed of light Uh-huh. Now under this guy Isaac Newton was saying that velocity add up that our U equals to U + VT u prime + VT or our U prime equals to U minus VT but under speed of light is constant. C= to C prime regardless of the frame you have speed of light is constant. So this is only true for everyday speed. as you approaching speed of light is constant.

[00:59:13]But when advertising concluded with this guy then the equation he used was able to even prove all the one that that is whereas the one that is did not prove for the speed of light. So we are able to conclude that if that statement is true for speed of light then it is what it is true for everything. So then when the law of physics is saying that you know there is a statement that say the law of physics is the same thing everywhere but for for the Newtonian physics is only the same thing when the velocity is far from speed of light but if we are now able to prove it that it's the same thing for speed of light then it's the same thing for everything. So now what am I saying is that sorry for taking it to 200 level. So now if we are saying this thing is true for a bigger number than the whole thing you have written already which is k + one then it's true for everything the condition given to you. Do you get what I'm saying? So you don't need to even know whether all the number given to you is like you don't need to know all the number since you already said the first case which include all your 1 + 2^ 4 + 3^ 4 plus dot + k^ 4. This already include everything assuming that K is a big number that comprise everything or that everything comprises whatever. Now you are now saying you want to now show it for let me just tell K like let me see a number after like at least there should be something at least there's life after life now isn't it? So if you able to show this then it is true you don't need to do anything again. So that's the powerful tool of mathematical induction. I'm just trying to just let you see like a glimpse of what induction is trying to show us. So now that's like because it can be asking y plus then we conclude that is true for the statement.

[01:01:01]That's just like an idea of it. So now let's open the bracket. So here we have but that doesn't make Newton to be Newton is to me new is still the greatest of all time because all he did was in 1964 rather there was no competition there. The man developed calculus that you are struggling with in math with k + one into what now we already have 2k² * 3k I've written this as k + 1 2k² * 3k >> 6k 4 2k² * 3k >> 6 k 6k * - I mean sorry 2k² * -1 >> right + 30 into if we expand this guy we have what kq + 3 k² + 3 k + what 1 bin write this thing three times you know you don't know what we did next are we together k + 1 k + 1 k + and multiply it together. You get this over what >> over 30 RIGHT WHAT ALL THIS ONE that we are doing right you know we've left the right hand side we are not touching it for now all this one just LHS write it on them left hand side starting from I think the last one before this write LHS on it don't write LHS on it you are doing another thing so that they know that what you are doing is the left hand side they look so now we have K + one.

[01:02:58]Which one?

[01:03:00]>> Show this one.

[01:03:03]Let's check. Let's check. This time this this time this this time this this okay it remain this K * 3 K square. Sorry.

[01:03:17]So here we have what? 3k * 3k 3k * - k + what?

[01:03:30]>> 30 k. Let's expand this plus what?

[01:03:34]90. You know if you expand if you expand this guy what kq + 3k² + what? 3k + 1.

[01:03:44]So if you multiply by 30, 30 * this we have 30k. 30 * this we have 90k² right?

[01:03:52]30 * this we have what >> 90k 30 * we have what 30 over what 30. Are we together?

[01:04:03]>> Huh?

[01:04:06]We are almost done with the left hand side. Just keep this one alone there since this one is already on the right hand side. So you don't need to touch it. Do we get? So just leave the k + one alone.

[01:04:16]So here we have we can write this directly without writing 30 and this first. You can write this directly without writing this. So here we have what? 6k 4 6 KQ + 3 KQ >> + 30 KQ >> 39 K >> 39 K Right - 2 K² + 3 K² that's K² + 90 K² >> + 91 K² and - K + 90 K >> 89 K plus what 30. Everything over what?

[01:04:57]>> So we are done with the left hand side.

[01:04:59]This is where the left hand side stops.

[01:05:02]>> Are we together?

[01:05:04]>> Can I clean this down?

[01:05:06]>> We've written it earlier. Now, now let's focus on the right hand side. Now the RHS.

[01:05:13]Now the R HS, what's the RHS? K + one, right?

[01:05:20]>> In what?

[01:05:21]>> K + 2. K + 2. Uh 2K + 3 2 K + 3 >> / what?

[01:05:39]>> Equals to 1.

[01:05:43]Now let's see this guy. K * 2 K * 2 K 2 K² K * 3 >> 2 * 2 K >> 2 * 3 are we done? Okay, we are done. 3k² + 9 k + 5.

[01:06:20]I hope it's not confused.

[01:06:23]Now can we proceed? Huh? I only expand this which is this. Then this one still remain. We can replace this with 7k.

[01:06:32]Isn't it 3k + 4k is 7k? Now so let's just let's write it directly.

[01:06:42]7 k + 6. Are we together? Can we proceed? So here we have k + 1 open bracket 2k² * 3k².

[01:06:57]>> It's looking all right. Thank God. 2k² * 9k >> 2k * 5.

[01:07:10]Now move to second one. 7k * 3k 21 k 7k * 9k >> 63 >> 63 k² 7k * 5.

[01:07:34]>> Now 6 * 3k >> 6 * 9 k 6 * 5 >> divided by what?

[01:07:51]>> 30. So equals to what? K + 1 into what?

[01:07:57]6 K 4 18 KQ + 21 KQ. That's 39 KQ.

[01:08:06]They're looking alike.

[01:08:10]Now let's move to square. 10k square + 63 k square >> + 18k square.

[01:08:18]>> It's 91. Thank god. 91 k². Let's move to k. Now >> 35k + 64k >> 89.

[01:08:31]>> 89 k plus what?

[01:08:34]>> Divided by what?

[01:08:37]>> Are they equal? Since the left you know we start right hand side since the left hand side is equal to the right hand side. So what happen the statement is true for what? For n as element of what natural number.

[01:09:01]So that's that for that.

[01:09:04]Do we get it?

[01:09:06]So there are cases that you don't need to expand. You can factoriize and there's a way you factor factor and both side will equal but you can't do it.

[01:09:14]This is like the quantity is like complicated. So you have to just expand both sides to make it equal.

[01:09:21]Most of the induction that I'll be giving to you they actually what they need is the proving.

[01:09:27]Are we done?

[01:09:30]>> So 3 minutes more.

[01:09:34]So mass is sweet now. No go for now.

[01:09:45]Yeah. This number two. We are just done with number two now. Yeah. Just done with number two.

[01:10:02]I've gone to sleep writing writing all those used to be very very vigilant Thank God for this refill. I don't buy plenty like this.

[01:10:42]It's actually 111 student that bought this for me.

[01:10:50]They know what I need. So they dash me ink.

[01:10:55]They cannot dash me car.

[01:11:00]It was one of the best key gift I've ever received. this this time I received gift most gift most was last year during my birthday December fact I was given this bag then this bag I was given that that I've not done it in my life before because it was very very wonderful They gave me wallets. They send money bag.

[01:11:38]Why they not give me drinks, food?

[01:11:43]Somebody cook fried rice like correct fried rice and she sliced meat inside with two big meat.

[01:11:54]I'm not exaggerating.

[01:11:57]How that birthday was was it was very very very people just sent 2K 2K 2K 2K. They were just sending me money.

[01:12:08]>> I can't wait to do another one.

[01:12:13]I can't wait to clean it.

[01:12:20]Very sweet now.

[01:12:25]So normally I don't used to celebrate sometime I'm doing birthday you nor yeah let's be fast are we done I'm not saying it so that if I'm doing can send me money I don't need to send money can Can I sing the body?

[01:12:56]My next birthday I'll be the one to I I will like I will do something for you.

[01:13:02]Money for day.

[01:13:05]Yeah. Yeah. Let's be less.

[01:13:08]There will be money then already. Yeah.

[01:13:10]Yeah. Money is money is not a problem.

[01:13:16]But we build something bigger than money.

[01:13:21]1 minute more. Let's be fast.

[01:13:39]Are we done? Are we?

[01:13:45]>> So in the course of still writing this, let me be telling us more story about Isaac Newton.

[01:13:52]Well, New really tried cuz he laid the foundation for modern physics. without new thing. Well, I can say advertising is actually a very crazy man like his brain is this word but Newton is still like because during his time there was no competition there 16 and all those achievement of Newton like colors you know Newton is the one who said like white light if you point white inside a prison is going to give like I think seven or eight different color like all those optics and everything is the one that all this calculus is the father of calculus. was the one that started calculus like he wanted to prove one of these things and mathematics could not do him right that's why he invented calculus so that's why invented calculus he did a lot of things and you know all those things that did he finish it at the age of I think 24 years or 25 he started around 20 or 21 years and I think he finished it around 24 24 years or so like somebody like Imagine me I don't can I still writing actually his brain is beyond this word ah that man like ah if you get level you know what I'm talking about is very sweet which mean any student Yeah, there is pure physics. Wow, you enjoy there is one there is one one course that equation 127 improving resion.

[01:15:52]Ah the last circumst show that prove that all those clust now we are done with that one. Now number three express sin of fifth power of theta as um as product of multiple Yeah.

[01:22:24]You see all the Okay, I think I Okay, So for signus I sin what? I actually let me just write it. If this one by the rationalizing rationaliz I sinus square like a + a minus. So if you use this multip so this guy give cos² - i² - 1 do we know that so 1 * 1 so this give us cos so i rational All it does again is only this value for complex numbers we add. If it is cosine we add it we subtract you subtracting because now = cos theta + i sin theta - cos theta whatus give us this cancel isn't itus cancelus plus I + I 2 I >> So - 1 = what 2 I do you get that part from the start we are given If this is this one is this why we need and that's the only way we can get we need this 90 that's the only recipro then subtracts instead of subtraction we do addition if we add this guy this one do we get but it's subtract am I don't get this is the only way you can now the Now is that since our question has a power of five. So we have to like we have to put this to power.

[01:26:59]Let me rewrite it. So this guy I s this we have to do same thing with this right so that the question will not change isn't it are we together are we together so this one become 2 i^ 5 what is that This this one 2^ 5 sin 5th power do we get to this part now we have to expand this guy and setal to this am I communicating and don't forget for the if Is this place don't get here?

[01:28:25]Es theta. So where = cos theta + i sin theta. Then 1 / give us the minus what I sin theta which I prove or I don't need to show the proof. So now give us what cos theta + i sin theta minus what cos theta minus what equals to what 2 i sin whatsus i have 2 i If - 1 / z = 2 i sin theta then z^ 1 2 i sin what n theta according to the are we together we are coming back to I just initial state now. So now um then how can we know at the end of the day something that we can write powertals to this right. So now how can we power by taking the power - 1 = 2 I so that we have the question inside then here we have - 1^ 2^ 5 I sin what power theta are we together so we have - 1 / z = 32 I do we know that = i² = i 4 * r = I² all² * I² * I - 1 square= what I soals sorry this are we together do we get to this part so we binomial to expand this binomial.

[01:31:33]So at the end of the day when we expand it now something like this for binomial not let you have a - b is what a power n a power n this factori - plus - n -1 a^ n - b² / 2 factorius n into n -1 n - 2 a power nus 3 bq Q / 3 factorial plus N into N -1 N you don't need this what you find also come down a^ nus 4 B what 4 factorius - N into N -1 N - 2 N - 3 A power n - 5 B^ 5 / what five 1 2 3 4 6 We've already goten um since our power is five. So let's stop here. Are we together?

[01:33:31]Now let's begin.

[01:33:33]Now our Z is like a now isn't it? Our 1 is like B. So to expand this guy, if you expand this guy, we have Z power n factorius n 5 into our a nus one is what? Four, right? What's our one? You don't need to put in minus again. the because of this minus that's why weus plus are together. So 1 / z factori 5 n - one 4 a z 1 square that's 1 square factor which isn't it plus the next one is 5 into 4 into 3 Z² 1 Z right plus then the next one + n * 4 * 3 * 2 a power 5 - 4 then b 1 / 4 / 4 factorial 4 factorial 24 then the last oneus 5 into 4 into 3 2 Z 5 - 5 is 0 is done right then we have B^ 5 1 / 5 rightid 5 don't forgetals what 32 I s what power let's write it don't write you don't need you can just expand and I will show us the simple way to expand I want it to be on a straight line so write like this right like this then Right.

[01:36:44]Right. All these are on a straight line.

[01:36:47]Together I power.

[01:37:08]Hello guys. Are we following?

[01:37:14]Are we following?

[01:37:18]Sharp. Sharp. Okay, now it's getting clear. I mean, it's clear.

[01:37:24]It's clear.

[01:37:34]Okay.

[01:37:58]All right. All right. You can just screenshot it. They writing it before I clean the board. So you this guy is serious. Somebody's there everywhere.

[01:38:19]I'm just playing with Are we done?

[01:38:47]I don't know. I don't This is actually one of the hardest question.

[01:39:33]Can I in terms of I don't think more than I was like, I feel like those things are true.

[01:40:28]I don't believe I I'm doing magic tell.

[01:41:13]Okay, don't worry. I will see. I will I will continue from this part. I believe no longer.

[01:41:47]mistake was even found to be true.

[01:41:55]Yeah, there was a time was like the earth should be either expanding or contracting but he couldn't was like but when he died many many Guys, can I slow right now? I don't know just so now listen I will start from here don't forget the initial condition = 2 + 2 I sin theta then n - 1 = what 2 I sin what n theta right so now here we have - 1 z -1 all power. Now let's expand again. We don't need to write it.

[01:43:39]We shown it earlier. Right? Now in binomial theorem plus one.

[01:43:54]So the next one will reduce by one. Then this one power one.

[01:44:01]The next one since it's minus plus minus plus the next one will also reduce by one then this one will carry square.

[01:44:14]Are we together?US plus minus Z will also be reduced by one. Then this one will carry Q plus Z will be reduced by one. Then this one will carry four power.

[01:44:33]I think there's one Z totally carry fifth power. Then we need the question.

[01:44:47]If you have a bus whatever power is what? If power what do you have? If the power is what do you have? The power is what do you have? 1.

[01:45:02]If the power is four what do you have? 1 4.

[01:45:05]If the power is five what do you have?

[01:45:09]1 10.

[01:45:18]>> So this is just don't need to stress.

[01:45:23]So that's the expansion of this.

[01:45:27]So this one will be reducing by one then this one will be increasing by one and this one has increased to the highest power then triangle to obtain the 1 equals to what 32 I X sin what this power theta which this one say it should be what I Yeah.

[01:46:21]This one will be reducing this first Z 1 Z.

[01:46:41]This one will be reducing. This one just like you have a minus 5 a5 then a4 a3 a square a1 a z then you have b 0 b1 b2 b3 b4 b5 do you get so that's what we did there now what are we going to do now let's gather all those together so here we have Z power 5 we've written this ear don't need to rewrite this z^ 5 bring this one to it 1 what zen can cancel out of it so we have - 5 z isn't it now let's also have no z can remove from this one which* 1. Am I right?

[01:47:52]Do we have this?

[01:47:56]Z.

[01:47:57]So here we have 5 Z 5.

[01:48:04]Then here we have Z this square. If you remove two from here, it remain 10 z agree.

[01:48:13]So this will be minus what? 10 * what? 1 / z right = 32 I sin power theta. Do we get that?

[01:48:28]Are we together?

[01:48:32]Let's talk now.

[01:48:33]>> Yes sir. together at all. So this guy give us 5 - 1.

[01:48:43]What is common to this?

[01:48:47]Bring it out. This one will be left with what? Zus plus. So don't mistake one + 10, right?

[01:49:04]Zus what = what? 32 I power then are we together?

[01:49:15]Now don't forget the 2in 2.

[01:49:26]So our so this guy is equal to what?

[01:50:20]want to make this guy. What do we do? We divide by 32 I right. So here we have sin^ theta = 2 i 2 i sin 5 - 10 I sin 3 + 20 I sin theta 32 IH So now sin you know since they have you can distin I will cancel.

[01:51:24]You have 32 which is what 10 I2 which is what 5in right 202 that's what 16 is going to be common are we together is it clear have sin powert theta - 5 sin theta plus what you have to inte which is very complex.

[01:52:25]You can use complex number to break down. It's very easy to complex number.

[01:52:36]So this is the answer. This guy fort 30 Now it's co plus plus the same answer 1 cos 5 + 5 sin cos theta + 10 cos it is it clear is it Yeah.

[01:53:15]So now let's write it.

[01:53:23]Let's write it.

[01:53:26]Is it clear guys?

[01:53:31]But I've explained the binomial expansion.

[01:53:37]Those of us that are watching from online did this thing.

[01:53:43]Okay. All right. I made this. You just come online like that. Let's try it so that we can start.

[01:53:57]You have to stretch your leg because all these are very simple to what we are about to do.

[01:54:18]The mistake I made is I forgot to bring Gary.

[01:54:23]That's a terrible mistake you can ever make.

[01:54:26]Not to have Gary at home or not to take Gary to where you want to take Gary.

[01:54:41]If you have every I have to open my phone.

[01:55:07]Medics not sh everything broadcast.

[01:55:29]>> Are you not seeing it there?

[01:56:00]I think it's clear. I think it's Why is he asking like I don't understand why is asking Can we see the board now? What of now?

[01:56:42]Can we see the board now? Now is it clear now Paul? Is it clear now?

[01:56:56]It is better. Okay.

[01:58:04]Mr. Are we ready?

[01:58:39]That's fresh.

[01:58:45]That's fresh.

[01:58:50]Even I'm afraid of me.

[01:59:28]That's always in the other room.

[01:59:47]I will.

[01:59:51]So, Everything is sterile.

[02:00:37]You see your face is very very very we are starting 102 very soon. No we have not started 102. We just want to give like we give like 5 minutes break.

[02:03:55]I'm coming. I quickly want to check WhatsApp.

[02:03:59]You guys just wait.

[02:04:02]>> I'm coming. Just give me a few minutes.