 Hello and welcome to the session. In this session we discuss the following question which says prove that if A is the arachnetic mean, G is the geometric mean and H is the harmonic mean for any two real positive quantities then first, A multiplied by H is equal to G squared. Second, A greater than equal to G is greater than equal to H. If we have any two positive quantities then the arachnetic mean denoted by say A would be equal to A plus B upon 2 then the geometric mean denoted by G is equal to square root of B the harmonic mean denoted by H would be equal to 2AB upon A plus B. Now let's see how do we get this harmonic mean H as 2AB upon A plus B because we have A capital H and B be in HP by this harmonic progression. This means this H would be the harmonic mean between A and B. Now as A, H and B are in HP so this means 1 upon A, 1 upon H, 1 upon B would be in this would be the arachnetic mean between 1 upon A and 1 upon B so 1 upon H would be equal to 1 upon A plus 1 upon B and this 1 upon 2 which would be equal to A plus B upon AB and this 1 upon 2 that is plus B upon 2AB and say that H is equal to 2AB upon A plus B and this H as you know is the harmonic mean between A and B. Now we get the harmonic mean H equal to 2AB upon A plus B. This is the key idea that we use in this question. Let's now move on to the solution. Now in the question we have that A is the arachnetic mean, G is the geometric mean and H is the harmonic mean for any two positive real quantities, positive quantities. Then as we are given that this A is the arachnetic mean that is the arachnetic mean between X and Y would be given by X plus Y upon which is the geometric mean would be given by square root of XY, harmonic mean would be given by 2XY upon X plus Y, multiplied by H is equal to G square, this is equal to A upon 2 multiplied by 2XY upon X plus Y. Now X plus Y, X plus Y cancels, 2 cancels with 2 and so this is equal to XY that is we have is equal to XY. You know that G is equal to square root of XY so this would mean that G square is equal to and we can write G square. Therefore we get A multiplied by H is equal to G square and this is what we were supposed to prove. So the first part of the question, now we are supposed to prove that equal to G is greater than equal to H compared to A minus G this would be equal to now as we know G is we have A minus G is equal to is equal to X plus Y minus and this one is greater than 0 that is it is positive is greater than 0 or you can say that A is greater than G. Now when we take the two quantities X and Y to be equal X is equal to Y this means we would be equal to that A would be equal to G. This is equal to Y, combining these two we can say that A is greater than equal to G and the equality X is equal to H is equal to G square. This means that A multiplied by H is equal to G multiplied by G. We also have that A is greater than equal to G so we can say that H would be less than equal to G or G is greater than equal to we can say that A is greater than equal to G and G is greater than equal to H and the equality X is equal to Y. This is what we were supposed to prove solution of this question.