Hello, students let us go through the next word problem based on Sn formula.

The sum is – 200 logs are stacked in the following manner, 20 logs in the bottom row, 19 in the next row, 18 in the next and so on. In how many rows are the 200 logs placed and how many logs are in the top row? Let’s go step by step. Let’s try to understand what is given. Log of woods something which you already know this is how it looks like. How are they placed, please understand. 20 logs are in the bottom row, right, so in the bottom row, in the first row, the first term there are 20 logs, right. In the second row there are 19, in the third row there are 18. So, a1 is 20, a2 is 19, a3 is 18. So here n represents the rows, first row n is equal to 1, second row n is equal to 2, third row n is equal to 3. And the terms are the number of logs, right. The question is how many rows, means find the total number of terms. So, we are supposed to find the value of n. A total of 200 logs are placed. How many rows are the 200 logs placed, so we are suppose to find the value of n, because number of rows means number of terms that is we are supposed to find the value of n, clear. So, we need to find the value of n as I said the number of rows and for that what is given to us, Sn is equal to 200 because the total 200 logs are placed, clear. The AP formed is 20, 19, 18 and so on as we discussed. These numbers form an AP with a is equal to 20, d is equal to 19 minus 20 which is equal to minus 1, right. And the total number of logs as we said is 200, so we know that Sn is equal to 200, very, very simple, right. Now, let’s solve. How do we solve? The total number of logs is equal to Sn is equal to 200. Sn is known to us, A is known to us, d is known to us. You are supposed to find n? How do we do it, we use the formula. What’s the formula, Sn is equal to n upon 2 into 2a plus n minus 1 into d, why we are using this formula, because d is known to us, right. Lets substitute, 200 is equal to n upon 2 into 2 into 20 plus n minus 1 into minus 1. Let’s move ahead, take 2 to the other side, it will get multiplied by 200. We get 200 into 2 is equal to n into 40 minus 1 into n is minus n and minus 1 into minus 1 is plus 1. Let’s move on, 200 into 2 is 400, n into 40 plus 1 is 41 minus n. So, 400 is equal to again open the bracket we get n into 41 that’s 41n and n into n is minus n square. Again we are getting a quadratic equation, let us solve it by the method of factorisation but before that write all the terms on one side. So, that there is zero on the other side. So, minus n square goes there, it becomes plus n square, 41n goes there becomes minus 41n and 400 remains there, so it is plus 400 is equal to zero. Let’s solve this quadratic equation by factorisation method. What is factorisation method? We need the factors of 400 which on adding gives us minus 41, why which on adding because 400 has got a plus sign, right. So, 400 factors which on adding gives us minus 41. Lets factorise 400, how do we do it, 2 into 2 into 2 into 2 into 5 into 5. What is the rule that we followed to find the factors, the biggest together and the smallest together, right. So 2 into 2 into 2 into 2 together that is 16, 5 into 5 together that is 25. Now 25 and 16, will it give 41 on adding. Yes, 25 plus 16 is 41 but then we want minus 41, so you want minus 16 and minus 25 because minus 25 and minus 16 is minus 41. So, let’s factorise it, n square minus 25n minus 16n plus 400 is equal to zero. Let’s take the common thing out n into n minus 25 minus 16. Because minus 16 is common the sign will change inside the bracket. So, inside the bracket, you will get n minus 25 and not plus 25, right, is equal to zero. Let’s move further, you get n minus 25 into n minus 16 is equal to zero. You get n minus 25 is equal to zero or you get n minus 16 is equal to zero, therefore n is equal to 25 or n is equal to 16. Here we are getting both the values of n as positive, now which one to chose and which one to leave. To decide that what we will do is, we will substitute n is equal to 25. We will substitute n is equal to 25 that is the number of logs, in the 25^th row, in a25 because with a25, a represents number of logs in a particular row, right. So, a25 is number of logs in the 25^th row. So, let’s find the number of logs in the 25^th row, a25 is plus 24d, right which is equal to 20 plus 24 into minus 1, right, which is 20 minus 24 which is equal to minus 4. When we take n is equal to 25, we get number of logs in that 25^th row as minus 4. But can number of logs be negative. The number of logs cannot be negative, right, minus 4 logs not possible. Since the number of logs cannot be negative, n is not equal to 25. So, what is n equal? N is equal to 16. Now, let’s find the number of logs in the 16^th row. That is we are supposed to find a16 because a represents the number of logs in the 16^th row, right. Let’s find a16 it’s a plus 15d that’s 20 plus 15 into minus 1, that’s 20 minus 15 which is equal to 5. So, the number of logs in the 16^th row, where 16^th is the top row because total number of rows are 16. So, a16 becomes the number of logs in the top row that is the 16^th row, right. So, number of logs in the 16^th row that is the top row is 5. So, the question is how many rows are there? There are 16 rows and how many logs are there in the top that is in the 16^th row, that is a16 which equal to 5, very, very simple, right.