 Hi and welcome to the session. Let us discuss the following question. The question says, find the value of n so that a to the power n plus 1 plus b to the power n plus 1 upon a to the power n plus b to the power n, with the geometric mean between a and b. Before solving this question, we should know that geometric mean positive numbers a and b, the number square root of a b. Let's now begin with the solution. In this question, we have to find the value of n so that a to the power n plus 1 plus b to the power n plus 1 upon a to the power n plus b to the power n is a geometric mean between a and b. We know that geometric mean that is g m between a and b is square root of a b since we have to find n so that a to the power n plus 1 plus b to the power n plus 1 upon a to the power n plus b to the power n is a geometric mean between a and b therefore a to the power n plus 1 plus b to the power n plus 1 upon a to the power n plus b to the power n is equal to square root of a b. Now this implies a to the power n plus 1 plus b to the power n plus 1 upon a to the power n plus b to the power n is equal to a to the power 1 by 2 into b to the power 1 by 2. On cross multiplying, we get a to the power n plus 1 plus b to the power n plus 1 is equal to a to the power 1 by 2 into b to the power 1 by 2 into a to the power n plus b to the power n. And this implies a to the power n plus 1 plus b to the power n plus 1 is equal to a to the power n plus 1 by 2 into b to the power 1 by 2 plus a to the power 1 by 2 into b to the power n plus 1 by 2. And this implies a to the power n plus 1 minus a to the power n plus 1 by 2 into b to the power 1 by 2 is equal to a to the power 1 by 2 into b to the power n plus 1 by 2 minus b to the power n plus 1. And this implies by taking a to the power n plus 1 by 2 common from this we get a to the power n plus 1 by 2 into a to the power 1 by 2 minus b to the power 1 by 2 is equal to by taking b to the power n plus 1 by 2 common from this we get b to the power n plus 1 by 2 into a to the power 1 by 2 minus b to the power 1 by 2. Now since a and b are different numbers on cancelling a to the power 1 by 2 minus b to the power 1 by 2 from both sides to the power n plus 1 by 2 is equal to b to the power n plus 1 by 2. Now this implies a to the power n plus 1 by 2 upon b to the power n plus 1 by 2 is equal to 1. And this implies a by b to the power n plus 1 by 2 is equal to now 1 can be written as a by b to the power 0 since base is same therefore we can compare powers so this implies n plus 1 by 2 is equal to 0 and this implies n is equal to minus 1 by 2. Hence the required value of n is minus 1 by 2. This is our required answer so this completes the session. Bye and take care.