 Hello and welcome to another problem-solving session on sequence and series We have been solving problems related to some of end terms of an AP so Let's take another problem in this series. It says how many terms of the series 54 51 48 dash So on and so forth be taken so that their sum is 513, right? And it seems there will be something called double answer So we'll see what is this double answer and we'll explore and evaluate that. So let's begin this solution. So What is the series? series here given is 54 51 48 so on and so forth, right? You can see all multiples of three. They are So, you know going forward it will be 9 6 3 0 Minus 3 like that. It will keep on going right? This is what is the series now? Clearly a is 54 and what is D? D is clearly minus 3 Right here. Lots of people make mistake D is nothing but 48 minus 51 rather than 51 minus 48 later term minus the preceding term, right? So minus 3 D is negative D is negative. So keep that in mind now. Oh, it says that the sum is 513. Let's say Sn is 513 Okay n terms and we have to find out this n and by the formula of first n terms of an AP we can write This is to a plus n minus 1 times D and this quantity has to be 513 Let's deploy the values of a and B and try to find out n. So N upon 2 twice a that means 2 times 54 Plus n minus 1 times minus 3 is equal to 513 right, I can take this 2 here to the other side and multiply it and it will become n times 1 not 4 1 not 8 rather plus 3 Right minus 3 into minus 1 is plus 3 minus 3 n is Twice 513, right? So this 2 comes here Multiplies, right? So this is n within brackets 1 1 1 minus 3 n is equal to 1 0 2 6 So this is nothing but n into thrice 37 minus n isn't it is equal to 1 0 2 6 so n Into 37 minus n will be 1 0 2 6 by 3 Right, so this is 37 n minus n square is equal to 3 and 4 and 2 yeah, so the equation is n square minus 37 n Plus 3 4 2 Is equal to 0. This is the equation which you get so clearly this is a quadratic equation And we have to now solve this so that we get the value of n either you go for 3 that are areas rule or splitting the middle term will also help so you can see This is nothing but 18 times 19 is 342 Right, so I can write n square minus 18 n minus 19 n Plus 3 4 2 Is 0 Okay, so clearly n times n minus 18 minus 19 times n minus 18 is 0 So splitting the middle term helped So hence let me write over here. So this becomes n minus 18 n minus 19 is 0 so that means either n is equal to 18 or n is equal to 19 so we get two terms 18 and 19 and hence they're asking To explain why these two terms to two ends right that means some of In the same AP some of first 18 terms is also same that is how much 513 and some of us 19 terms is also 513 This is happening because there is a negative D in this case. So if you notice 54 48 Sorry 51 was there in the middle. So 51 and Then 48 and then so on and so forth. These are all multiples of three if you notice. So this will be 9 6 3 0 Minus 3 minus 6 and so on and so forth now if you see some of this these many terms or Some of These many terms Same why because some there is one term which is 0. So addition of 0 will not Impact the sum so hence. Oh, sorry. I have to just This one till 3 not till 6 This right so some of these many terms Whatever is the value of n and the next term also if you're adding that is 0 So some is not getting impacted since there is a 0 over here Hence you can see there is a right the sum remains same for these if if you see till 3 these are 18 terms Okay, and till 19 till 0. These are 19 terms right Clear now it is not that only 18 terms and 19 terms will have the same sum if you see 17 term Let me draw like this. So these these are 17 terms so 17 and 20 terms also that is if you include in 17 term if you include these three terms as well three zero and minus three right that means add three more terms that is s20 They are all same because s 17 is till 54 51 Dot dot dot till 6 and s20 will be this plus three Plus zero Minus three which is again same some right isn't it so there will be lots of pairs of You know sums which are same for example again s16 if you look closely will be equal to s21 Similarly s of 15 15 terms will be equal to s of 22 and so on and so forth You could see you can check why because there are adding of you know three and minus three six and minus six So the sum is getting zero I hope this is understood till s17. We have to go to six We have one sum and then until s20 you add three zero and minus three which will get you the same sum Which is equal to some till 17 terms so some till 20 terms Include three zero and minus three So hence it is equal to some till 17 terms as well So hence you will get double answers You'll get two values of n for which the sum of n terms is same