Algebra

Geometrical Progression

General Form of a Series in G.P.

𝑎, 𝑎𝑟, 𝑎𝑟², 𝑎𝑟³, ⋯, 𝑎𝑟^𝑛−1 let 𝑎=first term, 𝑟=common ratio, 𝑙=last of 𝑛 terms, 𝑠=sum of 𝑛 terms; then 𝑙=𝑎𝑟^𝑛−1 𝑠=𝑎𝑟^𝑛−1𝑟−1 or 𝑎1−𝑟^𝑛1−𝑟 If 𝑟 be less than 1, and 𝑛 be infinite, 𝑠=𝑎1−𝑟, since 𝑟^𝑛=0

Proof

By multiplying the series by 𝑟, and subtracting one series from the other.

Geometric Mean

Geometric mean between 𝑎 and 𝑏 = 2𝑎𝑏

Sources and References

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