 Hello and welcome to the session. Let us understand the following question which says 200 logs are stacked in the following manner. 20 logs in the bottom row, 19 in the next row, 18 in the row next to it and so on. See figure 5.5. In how many rows are the 200 logs placed and how many logs are there in the top row? Now let us see the figure. There is the figure where there are 20 logs in the first row and then 19 in the next row, then 18 in the next to it and then 17 logs and so on. The number of logs in a row is 1 less than the number of logs in the preceding row. So in this row there are 19 logs and in this row there are 20 logs. So this row has 1 less than this row. So let us write the solution. The number of logs in the rows form an AP that is 20, then 19, 18, 17, so on. Here A is equal to 20, D is equal to 19 minus 20 which is equal to minus 1 and it is given to us as there are 200 logs. So SN is equal to 200. Therefore SN is equal to N by 2 multiplied by 2A plus N minus 1 multiplied by D which implies SN is equal to 200 is equal to N is equal to unknown. So we write here N alt is divided by 2 multiplied by 2 multiplied by A is equal to 20 plus N minus 1 D is equal to minus 1. Taking 2 on the left hand side which implies 400 is equal to N multiplied by 40 plus N minus 1 multiplied by minus 1 which implies 400 is equal to N multiplied by 40 minus N plus 1 which implies 400 is equal to N multiplied by 40 plus 1 is equal to 41 minus N which implies 400 is equal to 41 N minus N square which implies N square minus 41 N minus 400 is equal to 0. Now spreading the middle term we get N square minus 25 N minus 16 N plus 400 is equal to 0 which implies taking N common from this 2 term we get N multiplied by N minus 25 minus 16 common so we get here N minus 25 equal to 0 which implies N minus 16 multiplied by N minus 25 is equal to 0 which implies N minus 16 is equal to 0 or N minus 25 is equal to 0 which implies N is equal to 16 or N is equal to 25. Now case one when N is equal to 16 then number of logs in the 16th row is equal to the 16th term of an AP with first term that is A is equal to 20 and common difference D is equal to minus 1. Therefore number of logs in the 16th row is equal to A plus 15 D which is equal to 20 plus 15 multiplied by minus 1 which is equal to 20 minus 15 which is equal to 5. Now case two when N is equal to 25 then number of logs in the 25th row is equal to 25th term of AP whose first term is equal to 20 and common difference D is equal to minus 1. Therefore number of logs in the 25th row is equal to A plus 24 D which is equal to 20 plus 24 multiplied by minus 1 which is equal to 20 minus 24 which is equal to minus 4 and number of logs cannot be negative. Therefore N is not equal to 25 hence number of rows is equal to 16 and number of logs in top row is equal to 5 which is the required answer. I hope you understood the question that's all for the session. Bye and take care.