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Relationship between A.M, H.M & G.M

Raj Kumar
05/02/2018 0 0

If GM, AM and HM are the Geometric Mean, Arithmetic Mean and Harmonic Mean of two positive numbers respectively, then,

GM2 = AM × HM

Three numbers a, b and c are in AP if b=a+c2">b=a+c/2

Three non-zero numbers a, b and c are in HP if b=2aca+c">b=2ac/a+c

Three non-zero numbers a, b and c are in HP if a−bb−c=ac">ab/bc=ac

a−bb−c=ac">Let A, G and H be the AM, GM and HM between two distinct positive numbers. Then

(1) A > G > H
(2) A, G and H are in GP

a−bb−c=ac">If a series is both an AP and GP, all terms of the series will be equal. In other words, it will be a constant sequence.

a−bb−c=ac">Power Series : Important formulas

1+1+1+â?¯ n terms">1+1+1+â?¯ n terms

a−bb−c=ac">=∑1=n">=1=n

 

a−bb−c=ac"> 

1+2+3+â?¯+n">1+2+3+â?¯+n

a−bb−c=ac">=∑n=n(n+1)2">=n=n(n+1)2

 

a−bb−c=ac"> 

12+22+32+â?¯+n2">12+22+32+â?¯+n2

a−bb−c=ac">=∑n2=n(n+1)(2n+1)6">=n2=n(n+1)(2n+1)6

 

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