 Hi and welcome to the session. I am Asha and I am going to help you with the following question that says, if a plus bx upon a minus bx is equal to b plus cx upon b minus cx is equal to c plus dx upon c minus dx such that x is not equal to 0, then show that a, b, c and d are in gp. Let us now begin with the solution and we are given that a plus bx upon a minus bx is equal to b plus cx upon b minus cx is equal to c plus dx upon c minus dx. Now, first let us take the first two that is a plus bx upon a minus bx is equal to b plus cx upon b minus cx. Now, applying component row and dividend row on both the sides this can be written as a plus bx plus a minus bx upon a plus bx minus of a minus bx is equal to b plus cx plus b minus cx upon b plus cx minus of b minus cx or bx cancels out with minus bx and a with minus a, a cx with minus cx and b with minus b, we have 2a upon 2bx is equal to 2b upon 2cx. Now, 2 cancels out with 2 here 2 with 2 and we can write it as a upon b is equal to bx upon cx or a upon b is equal to b upon c. So, this implies a, b and c are in gp. Let us take the last two terms. So, we have b plus cx upon b minus cx is equal to c plus dx upon c minus dx. Now, here again applying component row and dividend row on both the sides this can further be written as b plus cx plus b minus cx upon b plus cx minus of b minus cx is equal to c plus dx plus c minus dx upon c plus dx upon minus of c minus dx. This further implies cx cancels out with minus cx, b with minus b, dx with minus dx and c with minus c. So, we have 2b upon 2cx is equal to 2c upon 2dx which can further be written as b upon c is equal to cx upon dx or b upon c is equal to c upon d. So, let this be equation number 2 and the first one be equation number 1. Now, from 1 and 2 we get that a upon b is equal to b upon c is equal to c upon d which further implies that a, b, c and d are in gp. So, this completes the solution take care and have a good day.