 Hi and welcome to the session. Let us discuss the following question. Question says, if the sum of first 7 terms of an AP is 49 and that of 17 terms is 289, find the sum of first n terms. First of all, let us understand the key idea to solve the given question. Sum of n terms of AP is equal to n upon 2 multiplied by 2a plus n minus 1 multiplied by d, where a is the first term of AP and d is the common difference. This is the key idea to solve the given question. Let us now start with the solution. We are given that sum of first 7 terms of AP is equal to 49. Now we know sum of n terms of AP is equal to n upon 2 multiplied by 2a plus n minus 1 multiplied by d, where a is the first term and d is the common difference. Now clearly we can see we have given sum of 7 terms. So, here we can write 49 is equal to value of n is equal to 7, 7 upon 2 multiplied by 2a plus 7 minus 1 multiplied by d. Now simplifying, we get 49 multiplied by 2 upon 7 is equal to 2a plus 60, multiplying both the sides of this expression by 2 upon 7, we get this expression and 7 minus 1 is equal to 6, so we can write it as 60. Now we will cancel common factor 7 on left hand side. We know 7 multiplied by 7 is equal to 49. Now we get the expression 14 is equal to 2a plus 60. Now dividing both the sides by 2 we get 7 is equal to a plus 3d or we can write a plus 3d is equal to 7. Let us name this expression as 1. Now next we are given in the question that sum of 17 terms is 289. Again we will use sn is equal to n upon 2 multiplied by 2a plus n minus 1 multiplied by d, here value of n is equal to 17 and value of sn is equal to 289. So we will write 289 is equal to 17 upon 2 multiplied by 2a plus 17 minus 1 multiplied by d. Now multiplying both the sides by 2 upon 17 we get 289 multiplied by 2 upon 17 is equal to 2a. We know 17 minus 1 is equal to 16, so we can write 2a plus 16d. Now we will cancel common factor 17 on left hand side. We know 17 multiplied by 17 is equal to 289. Now we get 34 is equal to 2a plus 16d. Now dividing both the sides by 2 we get 17 is equal to a plus 8d. Now we can write 17 is equal to a plus 3d plus 5d. 8d can be written as 3d plus 5d. Now we know a plus 3d is equal to 7. We have named this expression as 1. Now let us name this expression as 2. Now we will substitute value of a plus 3d from expression 1 in expression 2 and we get 17 is equal to 7 plus 5d. Now subtracting 7 from both the sides we get 10 is equal to 5d. Dividing both the sides by 5 we get 2 is equal to d. Now we can write d is equal to 2, so we get common difference is equal to 2. Now substituting value of d is equal to 2 in expression 1 we get a plus 3 multiplied by 2 is equal to 7. We know 3 multiplied by 2 is equal to 6, so we can write a plus 6 is equal to 7. Now subtracting 6 from both the sides we get a is equal to 1. We have to find sum of first n terms of a p. We know sum of n terms of a p is equal to n upon 2 multiplied by 2a plus n minus 1 multiplied by d where a is the first term of a p and d is the common difference. Now sn is equal to n upon 2 multiplied by 2 multiplied by 1 we know value of a is equal to 1 plus n minus 1 multiplied by 2. We know value of d is equal to 2. Now simplifying we get sn is equal to n upon 2 multiplied by 2 plus 2n minus 2. Multiplying these two brackets we get 2n minus 2. Now plus 2 and minus 2 will cancel each other and we will get sn is equal to n upon 2 multiplied by 2n. Now this can be written as n upon 2 multiplied by 2n. 2 and 2 will cancel each other and we get sn is equal to n square. So our required answer is sum of first n terms of a p is n square. This completes the session. Hope you understood the session. Take care and have a nice day.