 Hi and welcome to the session. I am Asha and I am going to help you with the following question that says if A, B, C, D, R, N, G, P prove that A raised to the power N plus B raised to the power N, comma B raised to the power N, plus C raised to the power N, comma C raised to the power N, plus D raised to the power N, R, N, G, P. Let us now begin with the solution. And here we are given that A, B, C, D, R, N, G, P implies B upon A is equal to C upon B is equal to D upon C and let these three ratios be equal to R. So let us implies that B is equal to AR, C is equal to VR and D is equal to CR. And we have to show to the power N, comma B raised to the power N, plus C raised to the power N, and C raised to the power N, plus D raised to the power N, R, N, G, P. That is we have to show that B raised to the power N, plus C raised to the power N, whole square, is equal to A raised to the power N, plus B raised to the power N, into C raised to the power N, to the power n plus d raised to the power n. If A, B and C are in GP, then B upon A is equal to C upon B or we can say that B square is equal to AC. So to show that A, B and C are in GP we will show that B square is equal to AC or in order to show that A raised to the power n plus B raised to the power n comma B raised to the power n plus C raised to the power n comma C raised to the power n plus d raised to the power n are in GP, we will have to show this condition. So first let us start with the left hand side. So the left hand side is equal to B raised to the power n plus C raised to the to the power n whole square. This is further equal to, now b is equal to a r so substituting a r raise to the power n plus c is equal to b r and b in turn is equal to a r so a r into r plus gives a r square. So we have a r square raise to the power n whole square which is further equal to a raise to the power n into r raise to the power n plus a raise to the power n into r raise to the power 2n whole square. This is further equal to a raise to the power 2n into r raise to the power 2n into 1 plus r raise to the power n whole square. So this is the value of left hand side. Now let us find the value of right hand side which is equal to a raised to the power n plus b raised to the power n into c raised to the power n plus d raised to the power n. Now a raised to the power n as it is plus v is equal to a r, so substituting a r raised to the power n place of b. Now c is v r and v in turn is equal to a r, so this is equal to a r square, so we have a r square whole raised to the power n plus d is equal to c r a r square, so a r square into r is a r cube, so here we have a r cube whole raised to the power n. So this is further equal to a raised to the power n into 1 plus r raised to the power n. Now from here again taking a raised to the power n common and r raised to the power 2n common here we have 1 plus r raised to the power n, so this implies a raised to the power 2n into r raised to the power 2n into 1 plus r raised to the power n whole square, so this is the value of right hand side. Now from 1 and 2 we find that both the values are equal, the value of left hand side is equal to the value of right hand side, so this implies left hand side is equal to right hand side or we have v raised to the power n plus c raised to the power n whole square is equal to a raised to the power n plus b raised to the power n into c raised to the power n plus d raised to the power n or v raised to the power n plus c raised to the power n upon a raised to the power n plus b raised to the power n is equal to c raised to the power n plus to the power N upon v raised to the power n plus c raised to the power n. This implies that a raised to power n plus v raised to power n, v raise to the power n plus c raised to the power n, and c raised to the power n, plus d raised to the power n are G p. So, this completes the session. Take care and have a good day.