Notes on Harmonic progression

Raj Kumar
23/08/2018 0 0
  • Let a, b and c form an H.P. Then 1/a, 1/b and 1/c form an A.P.

  • If a, b and c are in H.P. then 2/b = 1/a + 1/c,  which can be simplified as b = 2ac/(a+c) 

  • If a  and b are two non-zero numbers then the sequence a, H, b is a H.P.

  •     The n numbers H1, H2, ……,Hn are said to be harmonic means between a and b,
  • if a, H1, H2 ……, Hn, b are in H.P. i.e. if 1/a, 1/H1, 1/H2, ..., 1/Hn, 1/b are in A.P.
  • Let d be the common difference of the A.P., Then 1/b = 1/a + (n+1) d ⇒ d = a–b/(n+1)ab.
  • If a  andb  are two positive real numbers then A.M x H.M = G.M2

  • The relation between the three means is defined as A.M  >  G.M  >  H.M

  • If we need to find three numbers in a H.P. then they should be assumed as 1/a–d, 1/a, 1/a+d

  • Four convenient numbers in H.P. are 1/a–3d, 1/a–d, 1/a+d, 1/a+3d 

  • Five convenient numbers in H.P. are 1/a–2d, 1/a–d, 1/a, 1/a+d, 1/a+2d

 

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