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 Σⁿ_r=1 r² – 3 (n(n+1))/2 + n

⇒ 3 Σⁿ_r=1 r² = n³ + 3n(n+1)/2 – n

= n/2 (2n² + 3n + 1)

⇒ Σⁿ_r=1 r² = n(n+1)(2n+1)/6

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