What is the relationship between arithmetic mean, geometric mean & hormonic mean?
Hello dude. This is the fully explained relationship with derivations. Let AM = arithmetic mean, GM = geometric mean, and HM = harmonic mean. The relationship between the three is given by the formula AMHM=GM2 Below is the derivation of this relationship. Derivation of AM × HM = GM2 xAMy arithmetic progression Taking the common difference of arithmetic progression, AM−x=y−AM y=2AM−x Equation (1) xGMy geometric progression The common ratio of this geometric progression is xGM=yGM y=xGM2 Equation (2) Equate Equations (1) and (2) 2AM−x=xGM2 2xAM−x2=GM2 x2−2xAM=−GM2 Equation (3) xHMy harmonic progression x11HMy1 the reciprocal of each term will form an arithmetic progression The common difference is 1HM−x1=y1−1HM 2HM−x1=y1 xHM2x−HM=y1 y=xHM2x−HM Equation (4) Equate Equations (1) and (4) 2AM−x=xHM2x−HM (2AM−x)(2x−HM)=xHM 4xAM−2AMHM−2×2+xHM=xHM 4xAM−2AMHM−2×2=0 2xAM−AMHM−x2=0 x2−2xAM=−AMHM Equation (5) Equate Equations (3) and (5) −AMHM=−GM2 AMHM=GM2 Hope you got what you was looking for. Th
