 Hello and welcome to the session. In this session we discussed the following question which says, if A, B, C, D are in continued proportion, prove that A is to D is equal to A cubed is to B cubed. We know that when three quantities say A, B and C are in continued proportion, then we say that A is to B is equal to B is to C. This is the key idea that we use for this question. Let's move on to the solution now. We are given that A, B, C and D are in continued proportion. So therefore, this means that A upon B is equal to B upon C is equal to C upon D. We are supposed to prove that A is to D is equal to A cubed is to B cubed. That is, A upon D is equal to A cubed upon B cubed. First of all, we suppose let A upon B equal to B upon C is equal to C upon D B equal to K. Then this means that A is equal to BK, B is equal to CK and C is equal to DK. Now therefore, A is equal to in place of B we put CK. So it is CK into K that is equal to CK square and in place of C we put DK. So this is equal to DK into K square that is we get this is equal to DK cubed. So we can say that A is equal to DK cubed. Now next we have B is equal to CK and in place of C we put DK. So this is equal to DK into K that is DK square. Therefore we get that B is equal to DK square. Now first of all we would consider this LHS which is A upon D. So we have LHS is equal to A upon D. In place of A we put DK cubed. So we have DK cubed upon D. Now D cancels with D and we are left with K cubed. Thus we have the LHS is equal to K cubed. Next consider the RHS which is equal to A cubed upon B cubed. Now here in place of A we put DK cubed. So DK cubed the whole cube upon B cubed and in place of B we put DK square. So DK squared the whole cube. And so this is further equal to D cubed into K to the power 9 upon D cubed into K to the power 6. D cubed D cubed cancels and here we are left with K cubed. So we have the RHS is equal to K cubed. Now we got LHS also as K cubed and RHS also as K cubed. Thus we have the LHS is equal to the RHS that is A upon D is equal to A cubed upon B cubed. Hence we have proved that A is to D is equal to A cubed is to B cubed. This is what we were supposed to prove. So this completes the session. Hope you have understood the solution of this question.