Algebra

Harmonical Progression

General Form of a Series in H.P.

𝑎, 𝑏, 𝑐, 𝑑, ⋯ are in Harmonical Progression, when the reciprocals 1𝑎 ,1𝑏 ,1𝑐, 1𝑑, ⋯ are in Arithmetical Progression. Or when 𝑎:𝑏∷𝑎−𝑏:𝑏−𝑐 is the relation subsisting between any three consecutive terms. 𝑛^th term of the series = 𝑎𝑏(𝑛−1)𝑎−(𝑛−2) Approximate sum of 𝑛 terms of the Harmonical Progression, 1𝑎+𝑑, 𝑎𝑎+2𝑑, 𝑎𝑎+3𝑑, ⋯, when 𝑑 is small compared with 𝑎, =(𝑎+𝑑)^𝑛−𝑎^𝑛𝑑(𝑎+𝑑)^𝑛

Proof

By taking instead the Geometrical Progression, 1𝑎+𝑑+𝑎(𝑎+𝑑)²+𝑎²(𝑎+𝑑)³+⋯

Harmonic Mean

Harmonic mean between 𝑎 and 𝑏 = 2𝑎𝑏𝑎+𝑏

Sources and References

https://archive.org/details/synopsis-of-elementary-results-in-pure-and-applied-mathematics-pdfdrive