Relation between A.M., G.M. and H.M.

 
Let there are two numbers ‘a’ and ‘b’, a, b > 0
 
then AM = a+b/2
 
GM =√ab
 
HM =2ab/a+b
 
∴ AM × HM =a+b/2 × 2ab/a+b = ab = (√ab)2 = (GM)2
 
Note that these means are in G.P.
 
Hence AM.GM.HM follows the rules of G.P.
 
i.e. G.M. =√A.M. × H.M.
 
Now, let us see the difference between AM and GM
 
AM – GM =a+b/2 – √ab
 
             =(√a2)+(√b)–2√a√b/2
 
i.e. AM > GM
 
Similarly,
 
G.M. – H.M. = √ab –2ab/a+b
 
                 =√ab/a+b (√a – √b)2 > 0
 
So. GM > HM
 
Combining both results, we get
 
AM > GM > HM                                                   …….. (12)
 
All sequences of numbers cannot be put into A.P./G.P./H.P. Let us study these.
 

Important Points:

 
r3 (r – 1)3 = 3 r2 – 3r + 1
r = 1 : 13 – 0 = 3 . 12 – 3 . 1 + 1
r = 2 : 23 – 13 = 3 . 22 – 3 . 2 + 1
r = 3 : 33 – 23 = 3 . 32 – 3 . 3 + 1
r = n : n3 – (n–1)3 = 3.(n2) – 3(n) + 1
 
Adding
 
n3 = 3 (12 + 22 +…+ n2) –3 (1 + 2 + 3 +…+ n) + (1 + 1 +…+ n times)
 
n3 = 3 Σnr=1 r2 – 3 (n(n+1))/2 + n
 
⇒ 3 Σnr=1 r2 = n3 + 3n(n+1)/2 – n
 
        = n/2 (2n2 + 3n + 1)
 
 
 Σnr=1 r2 = n(n+1)(2n+1)/6
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