 Hey welcome friends to another session on sequence and series we have seen in the previous sessions what sequence and series are and we had started analyzing one type of sequences that is which can be represented as a you know or which in the nth term of which can be expressed as a function of n isn't it in the last session we saw an example related to that so I would request if you have not seen that it will be important to go through all those previous sessions to get a thorough understanding of whatever we are doing in this session now we have been given a you know sequence like that which is 25 10 17 26 and 37 and the idea is the objective in this session is to find out if we can figure out a generic formula for nth term of this sequence meaning thereby if let's say someone asks me to find out 100th term in this sequence then do I have a direct formula I don't want to really you know find out a trend over here and then go on finding out each and every element of this sequence and what we call as brute force method yes it is possible but it will take a lot of time unnecessarily and if someone asks me let's say 10000th term in this then it will be futile to even work it out on one by one basis so do I have a formula to represent any term in this sequence so that I just simply put the value of n over there and figure out the value of the term in that sequence that's the objective so let's try and analyze this sequence first of all so and in the previous sessions if you noticed we did difference of two consecutive terms and let's try to do the same thing here so if I find out the first and second term difference it is three right second term minus first term is three then here it is five and then again it is seven and then now it is nine and this is 11 and so on and so forth so did you notice something in the previous case we had all the difference so we I am calling it as layer one difference so the first difference you calculated for this you know from the sequence itself okay so these differences were constant in the previous case right where it was linear term but in this case it's not constant but if you go further down let's say take the second layer of difference so if you see this is two and this is clearly two again and this is two right and again another two like that so what do we observe we observe that the second layer of difference is constant layer two is constant right so it gives an indication that if you know the previous case was a linear term though in this case the linear or the term or any generic term can be expressed as a n squared plus bn plus c okay this is what is the understanding right there is a full-fledged proof around this particular thing but we are not going to discuss the proof here we are just trying to implement this what do I say listen to it carefully once again so if you have a sequence whose second layer of difference is coming out to be constant then that particular sequence any term can be expressed like this formula a quadratic function of n quadratic function of n right so Tn is a quadratic function of so remember we discussed that Tn is a function of n so in this case the function will be quadratic okay how we will look at it in different session right now let's apply it and try to find out a b and c and c that we are able to get a formula for this so now let us say n is equal to one so what is n is equal to one if you see n is equal to one from the sequence if you see this is two so let me write two and from from our formula if I deploy n is equal to one here in these two places I will get T1 as a into one square plus b into one plus c correct that is a plus b plus c right a plus b plus c is my first equation a plus b by c a plus b plus c and that equals two so I can write a plus b is equal to two please understand once again first term is to clearly know about it no doubt about it and I am assuming that this formula is going to work for every term that means if I deploy n is equal to one in this relationship I will get the nth term so if n is equal to one here put n is equal to one in this relationship you will get T1 so T1 from the given formula is a plus b plus c and from the actual value it is two so I can equate both of them so from formula it is a plus b plus c and from actual value it is two so let us try and let us call it equation number one let's go for n is equal to two if you see n is equal to two from the sequence the number is five is it so let us write five here five and from the formula what will it be it will be a into two square plus b into two plus c simply so again after simplification you can see this is four a plus b plus c is equal to five from the sequence the second term is five from the formula it is not only b to be sorry I missed a two here so this will be to be correct this is equation number two okay but we have three variables so we will not be able to solve for a b and c so let's go for the third n is equal to three n is equal to three and you can pick any n for that matter n equals to four five whatever the answer or the solution is not going to change so the third number was ten from the sequence if you see this is ten and from the formula we will get a into three square plus b into three plus c so this would imply nine a plus three b plus c is equal to ten right from the formula is the LHS from the actual value of the sequence is the RHS so this is equation number three now we have got three equation we can solve for a b and c we have learned that and find the value of a b and c so we will do this operation first two minus one so that the lone c gets eliminated so if you do this you will get three a plus b is equal to three this can be treated as equation number four and do this one three minus two right this will straight straight away eliminate c so this will give you five a plus b is equal to five ten minus five is five so this is equation number five now can I not eliminate b from four and five yes so do five minus four these two you'll get two a is equal to two so a becomes one b gets cancelled obviously same same coefficient so b gets cancelled a is equal to one so this is our result a is equal to one but we also need to find out b and c so if you use four from four from four what can we say b is equal to three minus three a correct and a was one so three minus three into one is zero right so b comes out to be zero and from there if you check equation number one equation number one was from one from one and I will write that equation one here a plus b plus c was equal to two so a is one b is zero so c is equal plus c is equal to two so what will c be c is equal to one is it it so hence my friends tn becomes an square right what was tn if you check see this was tn so a b and c I have got so it will be simply n square plus zero n plus c so tn is this was a n square plus bn plus c now deploy all the values which we found out a was one so it is n square one times n square plus zero times n plus one c was one so hence it is simply n square plus one okay let's check whether it is actually true so my general formula is coming out to be tn is equal to n square plus one let's go back to the sequence and see whether it is actually true so let me write the sequence here so we are getting tn is equal to n square plus one and now check let's say n is equal to one so tn from our formula is two very correct true n is equal to two so tn will be two square plus one which is five that's also correct so all this is so n is equal to three if you take tn will be from our formula three square plus one which is ten see all of them is matching so can't I find out hundredth term very easily so t100 will be simply what n square plus one right this is what the formula is so n is hundred so put hundred square plus one so that means ten thousand plus one that is one zero zero zero one ten thousand one okay so this is what is the hundredth term without even you know doing any calculation we can or for that matter any one by one operation and finding the next next next next like that instead of that directly I could figure out what is the value of t100 likewise for any any n whether it is one lakh one million one billion whatever you will be able to figure out the nth term in this so what did we learn in this session guys so hence if the second layer of the difference is constant then the nth term will be of this fashion right so keep this in mind so nth term will be a quadratic function if the first layer was constant it will be a linear function and in the next session we'll see that if the third layer is also constant right third layer is constant then it will be a cubic function so I hope you understood this