 Hi and welcome to the session. I am Asha and I am going to help you with the following question which says, if f is a function satisfying, f of x plus y is equal to fx into xy, for all xy belonging to n, such that f1 is equal to 3 and summation of fx, x running from 1 to n is 120, find the value of n. Let us now begin with the solution. And here we are given f1 is equal to 3 and summation of x running from 1 to n fx is equal to 120. So this implies f1 plus f2 plus f3 plus so on up to fn is equal to 120. Now f1 is given to us 3. Now let us find the value of f2, f3 up to fn. So f2 can be written as f of 1 plus 1 and this is equal to f of 1 into f of 1. Since we are given that f of x plus y is equal to f of x into f of y, for all x comma y belonging to n. So 1 into 1 plus 1. Both these numbers are natural numbers therefore it can be written as f1 into f1. Now f1 is 3 so we have 3 into 3 which is equal to 9. Now let us find the value of f3 similarly. Now f3 can be written as f of 1 plus 2. So this is equal to f of 1 into f of 2. Now f of 1 is 3 and f of 2 just we have found out this is 9. So this is equal to 3 2. Similarly f of 4 we can find out and this will be equal to 3 raised to the power 4 and so on fn will be equal to 3 raised to the power n. So f1 plus f2 plus f3 up to fn is equal to 120. Let this be equation number 1. Can be written as 3 plus 3 square plus 3 cube up to plus 3 raised to the power n is equal to 120. Since f1 is 3, f2 is 3 square, f3 is 3 cube, f4 is 3 raised to the power 4 and f of n is 3 raised to the power n. So you can see this is a symmetric progression whose first term is 3. Let us denote it by a and the common ratio is denoted by small rs, 3 which is greater than 1. Therefore the sum of this series is equal to 3 into 3 raised to the power n minus 1 upon 3 minus 1 is equal to 120. Since the sum of a geometric progression whose first term is a and the common ratio is r which is greater than 1, then the sum of this GP series is given by a into r raised to the power n minus 1 upon r minus 1. So this further implies 3 raised to the power n minus 1 is equal to 120 into 2 upon 3, 3 into 40 is 120. So we have 3 raised to the power n is equal to 40 into 2 is 80. 1 goes on the left-hand side so we have plus 1. So 3 raised to the power n is equal to 81 or we can say that 3 raised to the power n is equal to 3 raised to the power 4 and on comparing we find that n is equal to 4. Hence our answer is the value of n is 4. So this completes the solution. Take care and have a good day.