 sum of now we are going to find the way we did for sum of terms okay now it's interesting to observe something okay we'll try and observe that once we are done with this analysis now you remember how did we find some of n terms in an AP even remember what did we do you can unmute and say what did we do then remember yes or no reverse and add right what should we do here how should we find out some of n terms in a GP will reversing help in this case in this case reversing will not help okay so let's you know learn this process of finding reciprocate and add so you are saying reciprocate and add so s is equal to let's say 1 plus 2 plus 4 plus 16 plus 32 so you are saying will 1 by s be equal to 1 plus 1 by 2 plus 1 by 4 plus 1 by 16 plus 1 by 32 do you think this is correct right will this be helpful it is not going to help and anyways not correct so what do we do so learn this process very very crucial for all your efforts later on in sequence and series you need to learn this what is that let's say sn is equal to anyone and depicts number of number of right terms a1 a2 a3 a4 plus let's say a n a n minus 1 and a n sum of n terms of a GP okay this is step number 1 step number 2 can I write the same as n as a plus a r plus a r squared a r cubed plus like that a r n minus 2 and a r n minus 1 agree all of you agree to this one this part the same thing yes or no is step to clear all clear a common no no is clear is that clear to everyone now third step where the game starts I am multiplying both sides for the entire equation by common ratio r okay so what will happen this will become r times sn and the first term what is the first term after multiplying the right hand side what is the first term a r so I am not writing a r there I will write a r here okay then second term is a r squared so I'll write it here third term will be a r cubed I'll write it here plus here it will be a r n minus 2 plus a r n minus 1 and plus a r n agree do you agree to this how did n come because here it was r to the power of 0 from r to the power of 0 to n minus 1 if you multiply by r it will go from 1 to n correct is that okay I hope this is clear that it is going from 0 to n minus 1 if you multiply by r then it will start going from 1 to n clear yes is this clear now what you need to do is simply if this was 1 and 2 subtract 1 from 2 so you will get 4th sn common and r minus 1 right and here if you subtract all these terms will get cancelled 0 0 what is left left only will be a r n minus a correct so fifth sn is nothing but a common r to the power n minus 1 but isn't it below all of you agree so we just got sum of n terms of a GP correct in case of AP AP if you recollect what was in the case of AP sum of n terms or nothing but sn were up but our n upon 2 2 a plus n minus 1 d right here sn is a times r to the power n minus 1 by r minus 1 is this some clear how minus I subtracted 2 minus 1 right is this clear 2 minus 1 so hence when you subtract the RHS of second it will become minus right now right so please remember in GP we learned what all nth term a n is a r n minus 1 and now sn is a r n minus 1 divide by r to the power n minus 1 divided by r minus 1 here below so in a way you got something what is this sn can I write this as a so you got another identity if you check this is sn and this is how much a times 1 plus r plus r square plus dot dot dot till r n minus 1 yes or no this is what we got here if I cancel a from both sides what do I get r n minus 1 n is equal to r minus 1 times 1 plus r plus r square do you see something here do you see something here when n is equal to 2 you learned in ninth grade n is equal to 2 it will be r square minus 1 and this is simply r minus 1 r plus 1 or 1 plus r correct 1 to power n would be 1 yes so I'm just trying to express it like this right so it works for any the general general identity is this a to the power n minus b to the power n is how much a minus b then a n minus 1 plus a n minus 1 b plus a n minus 2 b square plus dot dot dot keep going a b n minus 2 and finally b n minus 1 this is the identity another piece of information for you right in this identity if you put a is equal to r and b is equal to 1 you will get this right so it's coming from there isn't it so hence you can find out the sum of this gp easily what is the sum of this gp r n minus 1 divided by r minus 1 is this clear the both a both both way analysis you got an identity from the sum of a gp okay always remember this is our very important identity and what else do you now see if a and b are integers then a to the power n minus b to the power n will always be divisible by a minus b interesting okay 19 to the power 5 minus 17 to the power 5 oh it's easy let's say 16 to the power 5 3 divide this yes or no you get that implications of same thing implications of all this right 3 divides 19 to power n minus 16 to power n whatever be the value of n for any value positive value of that means n is a natural number for any value of n interesting man so you must catch those insights correct right so you can see 38 to the power 45 minus 7 19 to the power 45 19 divides this yes or no 19 divides this 19 divides this entity correct so without calculation you can stay very clearly understood