 Hi and welcome to the session. I am Deepika here. Let's discuss the question. The houses of a row are numbered consecutively from 1 to 49. Show that there is a value of x such that the sum of the numbers of the houses preceding the house numbered x is equal to the sum of the numbers of the houses following it. Find this value of x hint sx minus 1 is equal to s49 minus sx. So, let's start the solution. We know that the sum of first n comes of an ap that is sn is equal to n by 2 into a plus l where a is the first term is the last term. So, this is a key idea behind this question. We will use the help of this key idea to solve our question. Solution. It is given in the question that in a row of houses 1 to 49, there is a one house number x such that the sum of the numbers of the houses preceding the house numbered x is equal to the sum of the numbers of the houses following it. Therefore, if x minus 1 x and x plus 1 so on till 49 are the number of houses, then the house number x in the question it is given, there is one house number x such that the sum of the numbers of the house preceding the house number x is equal to sum of the numbers of the houses following it. Therefore, house number till 1 to x minus 1 sum of the 1, 2, 3 plus so on till x minus 1 is equal to sum of the houses from x plus 1 to so on plus 49. This can be written in the form 1 plus 2 plus 3 plus so on plus x minus 1 is equal to 1 plus 2 plus 3 plus so on plus x plus x plus 1 plus so on till 49 minus 1 plus 2 plus 3 plus so on plus x act inside the sum of 1 plus 2 plus 3 plus so on plus x and minus 1 is equal to sum of the terms till x minus 1 is equal to n by 2 according to the key idea s n is equal to n minus 2 and in this case is x minus 1 by 2 into a plus last term a is our 1 plus last term is our x minus 1. So, this is equal to into x minus 1 by 2. Now, on the right hand side we have sum of 1, 2 plus 2 plus 3 plus so on plus x plus x plus 1 plus so on till 49. The sum is given by number of terms is 49, 49 by 2 into a is 1 plus l, l is here 49. So, this is equal to 49 by 2 into 50. This is equal to keep it as it is 49 into 50 by 2. So, on the right hand side the sum of 1 plus 2 plus 3 so on x is equal to is equal to n by 2. This is equal to x by 2 into 1 plus x. So, now we have substitute the values of these sums in this equation. We get x into x minus 1 by 2 is equal to 49 into 50 by 2 minus x into x plus 1 by 2. This implies x square minus x is equal to 49 into 50 minus x square plus x. This implies x square minus x plus x square plus x is equal to 49 into 50. This implies 2 x square is equal to 49 into 50. This implies x square is equal to 49 into 50 by 2. This implies x square is equal to 49 into 25 and x is equal to 7 into 5 which is equal to 35. The value of x satisfying the given condition which was there is a value of x such as the sum of the numbers of the houses preceding the house number x is equal to sum of the numbers of the houses following it is x is equal to 35. So, our answer for the above question is 35. I hope the question is clear to you. Bye and take care.