 start okay so first question on your screen so an AP has a property that the sum of first 10 terms is half the sum of next 10 terms if the second term is 13 then the common difference is the question take your time do not you don't need to rush through these questions cannot be solved just like that so people were just hitting it without thinking you're actually wasting your time here so 47 people have already solved this that's really surprising okay let's see so this is something which is okay so here was the question yes so it will take some time for you to solve this question so this was a question right so so a bigger canvas to solve now so what is the question AP has the property that the sum of sorry sorry some of the first 10 terms that means s10 the sum of first 10 terms okay first 10 terms is half the sum of next 10 terms s20 minus okay so s10 is equal to half s20 minus s10 did you understand this step guys all of you is this clear so this is the next next next 10 terms right so what do I get I get s10 is equal to half s20 minus s10 by 2 so that means 3 by 2 s10 is equal to s20 by 2 now you have to get the formula of s10 what is s10 s10 is 3 by 2 into 10 by 2 right into 2a plus 90 right you have to write it fast and then it is half into 20 by 2 that is 10 into 2a plus 19d I hope this is clear to everyone right are you following it so far so good go any problem this is I'm simply writing s10 and s20 now okay so what will this be so this is 5 so 15 by 2 let me write it here so 15 upon 2 into into 2a plus 90 2a plus 90 is equal to 5 times 2a plus 19d okay and one more relation was given second term is 13 what does this mean that means a plus d is 13 a plus d is 13 given right so we have two variables to equation we have to find out d that means we have to eliminate a is it so this 5 will be 3 and so hence it will be 6a plus 27d is equal to 4a plus 38d okay and I know a is equal to 13 minus d right or what you can do you can see from size more it is 2a and what is this 2a is equal to 11d get this one what Sneha what happened sorry Sneha can you unmute and say where is 3 by 2 what is 3 by 2 2 went to the yes so I have not mentioned that that you will have to do faster right this 2 will go to this side yep so I'm I will not write all the steps here so 2a is 11 11d so hence a is simply 13 minus d is equal to 11d so 26 minus 2d is equal to 11d so d is equal to 2 right but most of you I think you could not complete this on time so this was the first question guys okay so let's go back to yeah and let me delete this okay now next question okay so here is the first super fast in eight nanoseconds not a not an eight milliseconds you could solve this my god this is really surprising okay I took at least 45 seconds to solve this one anyways good if you are able to do it next yes someone's mic is on mute please put it on mute Anisha is to be done this question should have answered should have been answered in three seconds this question could have been solved in just three seconds so if you are aware of if you are formula based person then it will take 1000 years for you otherwise very very less amount of time is needed for this good most of you have got it correct that's nice but time should have been lesser I believe in this case so very easy why is it easy is because uh this particular question here 197 94 91 88 none of them are multiples of three okay so this is every number is just one next to multiple of three so it will it will go down and there will be numbers like four one right if you see this is 99 plus one this is 96 plus one this is 93 plus one so multiples of number multiples of three plus one so the next term will be minus two there will not be any zero right so after one that we see directly minus there will not be any zero so hence none of these right is that okay understood by all okay here give me why if all of all of you understood okay very good next question please here is the next question before that who's Nemo again there's someone every time there is someone who's Nemo this guy Nemo Nemo at least you should have put Nemo we could have related to you okay next question take your time this cannot be solved by just randomly picking something okay so last five seconds I hope this was pretty much simple yep very good simple simple problem it was but still lots of oh that's oh so such a why how many of you could not solve this question okay so lots of people could not solve this question okay they go so if abc been the arithmetic progression so all if abc is an arithmetic progression that means what to be is equal to a plus c is it now what is this term this term is a plus a plus c minus c what is this term a plus c minus a and here it is this a plus this c can be written as 2b simply and 2b right yeah that's what so hence you'll get it so c gets cancelled and a gets cancelled and this is nothing but 2a into 2c into 4b that's it so 8 and 2 4 4 16 so 16 abc three lying answer is it okay now tell me understood or not everybody got it just a minute just give me a second yeah sorry guys I was in some okay so yeah is this clear to everybody tell me the last question so simply you have to do what uh this so a this 2b was b plus c oh sorry a plus c so I wrote that then this 2b again I wrote a plus c and then I clubbed this a plus c as 2b so that gave me 4b here right so that's okay so next question so most of the questions will not require that much of uh writing okay also shuffling nana okay great next question on your screen guys a and c are same is it very simple question again you don't need to go to formulas and all that without formula basic knowledge of ap can solve this question so most of you have got it that's nice and uh for those who didn't get it so this is how it has to be how to solve this one very simple question again so what is to be found or common difference right what is common difference that we delete this one and if you see a 2 minus a 1 is equal to d and a 3 or a 4 a 4 minus a 2 this is also d a 6 minus a 5 this is also d right like that keep going 100 100 times again you will get d so add all what will you get you will get s 1 minus s 2 right is equal to 100 d right so d is equal to s 1 minus s 2 100 right so no formula so far I haven't used by 50 times there are 100 100 100 terms are there right so both are 100 100 200 terms were there so 100 100 each correct got it clear aren't so there will be 100 pairs and hence the sum d is this much did you understand see so in this question did we use any formula of some or whatever nth term and all that is this is clear to all just give me a clarity why acceleration not clear shall I explain again once again once again see what I'm saying is this is s 1 and this is s 2 now a 2 minus a 1 is simply d or what you do is you simply do s 1 minus s 2 do this s 1 minus s 2 what will you get you will get a 2 minus a 1 plus a 4 minus a 3 plus keep on doing and there will be such 100 pairs 100 such pairs oh actually that okay I thought okay never mind so this is d another way of doing it 100 times so hence 100 d is s 1 minus s 2 so d is s 1 minus s 2 upon 100 this one okay next sum of n terms of an AP is 2 n plus 3 n square which term of this AP is equal to 2 9 9 I hope this was easier let's view the distribution 25 oh my god only 15 why what happened in this what was the problem this was a very fairly simple and straightforward formula based time why okay let's see why do you take so much of time so there was a question and now sum of terms of an AP is this which term of AP is equal to 2 9 9 oh sorry so which term of this AP is equal to 2 9 9 now is equal to 2 n plus 3 n square okay you have to use this s t n is equal to s n minus s n minus 1 isn't it now you don't need to do all such things this is straightforward t n is equal to s n minus s minus 1 isn't it have we not discussed this if s n is given t n is s n minus s n minus 1 yep so hence now so t n n is what n you have to find out n you have to find out and it is given to be 2 9 9 let's say and what is n uh s n is 3 n square plus 2 n and subtract 3 n minus 1 whole square minus 2 n minus 1 isn't it this is 2 9 9 you have to just find out n isn't it so hence this is 3 n square plus 2 n minus 3 n square plus 2 3 just 6 n and if you expand it so minus 3 and minus 2 n plus 2 right you don't even get a quadratic right see 2 n 2 n gets cancelled so what is left 6 x 6 n so 2 9 9 is equal to 6 n and minus 1 so 6 n is 300 so n is 50 right why proud of haven't we discussed this that if s n is given and if you are asked to find out t n so this is this should be the approach rather than calculating a and d that will take a long time i think most of you would have done the same okay right so this should be understood by all so how much time did it take how much time did it take hardly if you know the approach hardly 30 seconds so that comes by practice okay so let's go to so now question guys here is the next question on the board some of us 10 terms of an ap is four times the sum of first five terms of the four there are two two times this this is useless one of the five terms can you you can ignore it has been written two times the four times the sum of first five terms that's it then the ratio of the first term to the common difference is this should be easy i believe most of you should have done this formula based direct okay so 25 got correct rest of them not correct okay so this was the question sum of first 10 terms of an ap is four times the sum of first five terms so direct formula will apply sum of first 10 terms is how much let n ad and all that be there so sum of first 10 terms is 10 by 2 into 2 a plus 90 okay and is four times the sum of first five terms so again five by two two a plus four d that's it so right so there should not be big deal here so this is five this five and this five will go so this is two only two left so two a plus nine d is equal to four a plus eight d okay so hence you will get what d is equal to two a right and first term to the common difference right so the first term is equal to d by 2 right so first term is double of the uh yeah so the common difference is double of the first term so hence the ratio is 1 is 2 no it's correct first term is half of the common difference right so basically first term is what a and in terms of a uh wait a minute uh first term to the common difference no it's yeah d is twice of a twice of a that's correct yep that's correct twice of a so hence a and two two a these are the terms a and d right so you have to find a is to d so a and a is to two a so one is to two right so a so most of you have got confused here i believe hence next so hence enough next question on god guys this is a sequence but it's not our ap bit of knowledge of previous grade will also be there that will be applicable here last 10 seconds okay so let's see uh still people are not very confident and still this is not that great okay so this one was very very simple so none of the questions will require any big big formula or something like that yeah you don't need to multiply and all it was simply some of first hundred terms if you didn't notice it some of first hundred terms it was simply so you see what is this this is nothing but one minus two one plus two isn't it or yes then what this is nothing but this one is three minus four three plus four right and similarly last one will be 99 minus 100 and 99 plus 100 so all of these are ones minus one in fact minus one minus one right so hence hence what is this this is nothing but minus one into one plus two plus three plus four this is not the gauss's formula i don't know so five plus dot dot 300 simple this and you know and you know this is 50 50 okay so minus yeah gauss's formula is this s is though the thing which we do in our formula and s is 100 plus 99 plus dot dot dot is still 100 and then you sum them up 2s is equal to 101 into 50 101 time 100 sorry right so s is 50 50 yeah okay so if you're referring to this then yes but yeah so here did you understand how to go about it most of you would have got panicked by seeing these two over here but that was the indication this is where you had to once again see once again i paired them up all of them this was the trick that's it so if you see one one square minus two square is one minus two one plus two a square minus b square form plus three minus four three plus four likewise all the terms will be in the last term will be 99 minus 100 times 99 plus 100 right so this is minus one so minus one times one plus two then minus one times three plus four then minus one times you'll get five plus six and so on and so forth minus one times 99 plus 100 right so hence minus one common and you'll get one plus two plus three plus 100 correct understood by all again we are not using any out of the world formula or something like that okay not even the basic formula which have been which have been which we have been practicing okay so so there's a message in all these questions i hope you're getting it how did you get minus one one minus two is how much are in one square minus two square is one minus two times one plus two so one minus two is how much minus one right but it hello any difficulty yes no but it very next question mute it mute it mute mute please it's of a triangle r in ap if cosine of one of the angles is zero then find some of the squares of the sign of other two angles it could have been solved by just taking an example very famous pathogen triplet and done did you guys use that so at least majority of you are getting it correct but then what is what does this mean what does this mean if i remove this what does this mean one of the angle is zero meaning what cosine of one angle is zero yes three four five you can use that money correct so that means is a right angle triangle and sides are in ap so what three four five right you can do that yeah you can use but i will show you how to do it to ap method as well so just in case if this is a competition what will you do so sine of one angle is how much three by five so square it and sine of the other angle is four by five square it you'll get one the famous one hardly 10 seconds hardly 10 seconds but if you wanted to solve it through ap then these sides are a minus d a and a plus d after ascertaining that this is 90 degrees let the sides be a minus d a a plus d okay so what is this so this will obey a plus d will be the highest so this is a high hypotenuse a minus d whole square plus a square correct so this is nothing but a square plus d square okay a square plus d square plus two ad is equal to a square plus d square minus two ad plus a square okay so this a square and this a square will go this d square and this d square will go so what do i get i get a square is equal to four ad that means a and a will go so a a is equal to four ad okay so a plus d is the hypotenuse so the sine of the two angles are nothing but a by four d sorry a by five d oh no uh yeah a by five d and a minus d is how much uh three d by five d is it sorry am i what wait a minute a is equal to four ad so the sides are what are the sides guys sides are uh four ad my three d four ad and five ad so instead of d so instead of three four five i'm getting these so the same again you'll come back to the same thing right so these are this was the answer so one is the answer right all of you understood give me a thumbs up why good so close this and go to the next question so some shuffling so let's go to the next question guys last five seconds let me see how did you so no direct question i told you there will be okay how many of you could solve this can you just post your response in the chat how many of you could solve this mera nyanesh front of stress four people 18 people are there nandini okay who else five people only and how come 18 there is a reason why i'm asking this so how come there are 18 people there okay so this was uh this was intended to be little you know how many if you could not solve it let's see okay so obviously this has something to do with ap right and a1 a2 is there so that was the catch that was the catch okay let's say this is a1 a2 a3 are in ap and let's say the common difference is d answer is a n minus one a1 an okay let's say the common difference is d okay so don't you think d is equal to a2 minus a1 this is the so if you are aware of this basic property of ap you will be able to do this okay so i have to find out this sum so this is what i am doing c d by d multiply this whole d by d okay so what will happen can i say one by d into take the numerator d inside so what will happen d by a1 a2 d by a2 a3 and so on and so forth d by an minus one an are you convinced is it okay can i not is it okay understand by all right just multiply divide by d so can i not write this as one by d and what is this this d i can write as a2 minus a1 by a1 a2 okay and this second d can be written as a3 minus a2 by a2 a3 do you get some clue here finally an minus an minus one by an minus one an right now why did i do this i'm now writing here if you see this will become one upon d and if you see this one if you break this open what is this this is nothing but one by a2 minus one by a1 for the other way round other way round one by a2 am i right one by a2 sorry a1 minus one by a2 correct the second term will be one by a2 minus one by a3 right so third term will be one by a3 minus one by a4 likewise it will continue and it will be one by an minus one minus one by an isn't it what do we see all getting cancelled except the first and the last term so we are getting one upon d correct and what we will be getting an minus a1 by a1 an right now what is the relation between an and a1 an is simply n minus one sorry n minus one d okay so an minus a1 an minus a1 by n minus one is equal to d right so hence or or you can say this is this implies let me write it here because i have small space so i can say one by d one by d is equal to n minus one by an minus a1 okay now put this here at this location so what will you get you will get n minus one by a1 n minus one by sorry this one n minus one by an minus a1 into an minus a1 by a1 an i am writing this here right i just replace one by d by this relationship so what will happen this will go so n minus one by a1 an below did you get it so what was the catch here can you explain again yes i can so let me do one thing where do i explain i need some bigger space okay so let me do it once again here only or so i'll delete all this because i don't have much space here so i'll delete everything once again they go once again so what i need was i multiplied and divide this entire thing by d why because this will become this will become one by d into d upon a1 a2 plus d upon a2 a3 plus so on and so forth d upon an minus one an okay clear this step is okay now replace all the d's i not replace this this is equal to sorry one by d into the first term d can be written as a2 minus a1 by a1 a2 second term can be written as a3 minus a2 all are d only right by a2 a3 finally an minus an minus one by an an minus one right why multiply by d by d so that's what i'm explaining so because i could write this a2 minus a1 so once the d gets in here this d gets in here i can write all these as respective difference of the denominators can you see that a2 minus this d here is a2 minus a1 and this d i'm writing as a3 minus a2 and likewise all so what how does it help because now if you see this becomes one by d into within brackets if you see the first look at this term this is nothing but one by a1 minus one by a2 yes or no if you look at this term this will become one by a2 minus one upon a3 likewise it will keep on going and this last term will become one upon an minus one minus one upon an and if you see these all will get cancelled one by one one by one all will get cancelled and only first time and last term will be left right so what is left one by d into and you take the LCM an minus a1 by a1 an this is left yes or no so what is an minus a1 an is equal to a1 first term plus n minus one d right so an minus a1 is n minus one d right put it here in this what will you get one upon d into an minus a1 i'm writing as n minus one d divided by a1 an so this d and this d goes so hence answer is a understood okay very good let's go to very yeah very good nice okay next am means arithmetic mean if a to the power n plus one plus b to the power n plus one divided by an plus b n is the am between a and b a is not equal to b then n is equal to what yes it was very easy only the only by look and feel it appears to be little deadly but otherwise yes or no easy no how many could do this already am method should be used scam methods scam method is what is scam method you should have scammed this question how to do that good most of you could do that yes you could simply put the values and game over so you should have yep so if you put a is equal to if n is equal to zero works see if you put n equals to zero it will become a plus b by one plus one a plus b by two is the am so this is scam method in competition use this method also these are low hanging fruits but for concept if you really have to find out so what is this this is nothing but a n plus one let's say if it is a subjective question you never know it can become a subjective question the test could become a subjective test a n plus b n is nothing but a plus b by two this is am right and then multiply both cross multiply so you'll get two a n plus one plus two b n plus one is equal to a n plus one and b times a n plus one plus a times b n plus one plus b n plus one right if you do the cross multiplication so this and this can be removed right yes mere bhai faster so i told you how to do this but then you must also know the legitimate way of doing it right that was the explanation i gave that tomorrow if this question is asked in the right what oh by two also will be there yeah sorry sorry sorry so by two also will be there i can't read this this this and this thanks for highlighting so this is two times this and oh no it's it was correct only why hey use cami unnecessarily right so it was okay only where was the problem below a n times a is a n plus one a n times b is b a n oh sorry not this was the issue this i can't understand this is sorry yeah okay now now what so you can cancel this no cancel this two and this right so hence now take a n plus one minus b a n and what uh is equal to b n a minus b n plus one okay so here i can take a n common and you'll get a minus b here you'll take b n common again you'll get a minus b so clearly a is not equal to b so you can cancel these two out right so you'll get a by b whole to the power n is one right so that means this is nothing but a by b whole to the power zero right so hence n is equal to zero this is a legitimate way of doing it okay so use of ap so both ways you could have tried shall next question good one nice shall next question guys our odd number of terms in an ap the ratio of the sum of the even numbered terms even numbered terms means second fourth sixth like that okay so you have solved it difficult oh my god wide distribution difficult how many could not lot of actually yeah okay so let me show you this question this was again let's say let us define the ap first so what are the ap's ap is a one sorry or let's say a a plus d a plus 2d a plus 3d and so on so there are odd number of terms so can i say there are let's say there are two n plus one terms two n plus one is odd let's say there are two n plus one terms so it will go up to a plus two and d am i right this is the ap is this an odd ap odd number of terms are there how many terms are there two n plus one is this this step is clear to everyone get give me a thumbs up so these are odd number one two three four count it there are two n plus one terms so odd odd number of terms clear yes yep now i have to first find out what ratio of the sum of the even numbered terms even numbered terms are this this this like that and the last one is not even number this is the odd number term two n plus one is term right so what is the even number so i'm writing se is equal to first term is a plus d second term is a plus 3d and so on and so forth and last term is a plus how many uh two n minus one d am i right am i right how many terms are there in even how many terms are these how many terms can you tell me so in two n plus one how many terms will be even terms even numbered terms so two four six eight how many no n number are even numbered n n are even terms and n plus one are odd terms so let's say if there are 101 terms in an ap 50 will be even and 51 will be odd getting it right 50 will be even terms even numbered terms and 51 will be odd number term odd numbered terms are you are you okay with this statement here if there are 101 terms in an ap so one first term third term fifth sum till hundred first term these are 15 numbers 51 sorry 51 in numbers and second fourth sixth still hundred is 15 number got it so hence we have n terms here so what is se by the definition first term is a plus d number of terms is n so can't you write it is n by two into twice a what is twice a two a plus two d right this is a twice a first term plus how many terms are there and mine and so n minus one into common difference what is the common difference between these what is the common difference in the first tape common differences can you tell me common difference 2d right so this is my se this is se so let's simplify it a bit so it is n by two and two a plus 2d plus 2nd minus 2d okay so this minus 2d plus 2d will go and there will be one two which will get cancelled so it is n a plus d am i right below a plus n a plus nd right this is for even number of terms let's see odd so odd terms are s o is odd terms are a plus a plus 2d plus a plus 4d plus one and so forth a plus 2nd how many terms n plus one terms are there correct so s o is nothing but n plus one terms by two twice a plus n plus one terms minus one times isn't it hence it is n plus one by two here you'll get two a plus nd right did i make some mistake two a plus oh sorry this is yeah it's correct only but i have to get um oh yes yes that was the mistake correct so so yeah it should be 2d here right yeah so hence what is it this is n plus one by two and two two will get cancelled so again you will get a plus nd right is it okay s o what was what was the question question was to find out s e by s o so below what will that be n by so n into a plus nd divide by this was s o s e sorry and s o is n plus one a plus nd so this two will go okay yes yes you could have you could have eliminated these two this is not possible why can you tell me why why is this not possible this is also not possible that's not wrong if you really i made it you know if you do very fast yes so here these are eliminated so you could you have choices between these two okay so anyways let's see okay so the last question of the evening guys last question when Anish says easy people are not able to get in okay good so easy for Anish chalo this was super super duper easy again we have to just work it out okay so what is this p term a1 to this thing directly write what p by two into two a one plus p minus one d right divided by q by two into two a one plus q minus one d right is equal to p square by q square right so this q and this q will go this p and this p will go this two and this two will also go so hence cross for deploy you'll get q times two a one plus q times p minus one d is equal to p times two a one plus q minus one d so lots of terms are going to be cancelled okay so this is two a one q plus q square what happened Anish something is there six sq how you can do that as well yep so i'm just i'm just doing this yeah let me complete and then we'll talk so q square pd so q square pd then q square minus q square d is equal to two p a one sorry why is this two where is this two no sorry but yeah two is there two is there two is there sorry two p a one plus p i'm missing something out here what is missing here two p times i did i made a mistake wait a minute uh two a q q p minus one and yeah that's no mistake correct so this is p q why did i get q square here why is it q into p minus one d ah why oh i had already done that on and again multiplied so that's what the mistake is so i had already done that i have already multiplied by q thanks for highlighting so now in in this case i have not done so hence it is two a q one two a one q sorry plus q pd minus qd right here it will be two a one p plus p qd and minus pd so p qd p qd will go so hence if you get commons two a one q minus p is equal to b common b common and it will be q minus p right so q minus p q minus p will go so d is equal to two a one okay now what you need to find out six and 21st term six term is a one plus five d 21st term is a one plus 20d now d is two a one so put two a one so a one a one plus 10 a one and a one plus 40 a one so hence 11 by 41 right this is one method what anish is doing is that's another method is this method clear to everyone so you don't need to find a and d you will not be able to find a and d sesh you'll only get a ratio between a and d right so 11 by 41 now what anish is saying is also right so did you follow this method this is a clear to everyone so simply add use the formula of sum of a piece right what anish did is this that if you have to find out sixth term right where is sixth term so sixth term is nothing but s6 minus s5 right s6 minus s5 and 21st term is s20 minus s21 right this is what sorry s 21st term is s21 minus s20 this is what you did anish is it so hence he put p is equal to but then how do you find out s6 minus s5 divided by s21 minus s20 you have to do this now tell me how did you find this out s6 minus s5 divided by s21 minus s20 so you'll only get you can only get s6 by s21 so I assume that the fraction is in least form so how does it help how does it help as if s6 minus s5 as divided by s21 minus s20 you have to do p is s p is okay like that you're saying okay you think sp divided by sq is equal to p square by q square like that and hence you're saying sp is equal to this is what you did right q square sp is equal to p square sq correct isn't it right so hence you found out but then that will give you again okay so you are saying s21 by s or s6 by s21 is equal to 6 square that is 36 by 441 this is what you're saying this is what you're saying you will get s6 by s21 from here but how do you get t6 by t21 that will that is a question t6 I assume fraction is the sum of power terms is p squared no that that's an assumption that's in one case it worked here but you need not work everywhere it worked here that because you assume that one that that case but I think you should go like this okay anyways because the time is short so let me wrap it up we'll discuss maybe in the class okay so here is the final final oh so lots of people but anyways the tally doesn't change much good so Naina is first Pranav, Smythi, Shreyas and who else it's he didn't Daksh I believe then Aden then who is this Monish, Alisha, Sneha, CHK people are clubbing up or what all of you are together huh great nice good so nice yeah keep it up I hope you like the questions the questions were not that straightforward they will not be asked in your board exams so don't worry but they will definitely be asked in various competitive exams okay so so fair enough see you again next week bye bye take care this is good night