Algebra

Arithmetical Progression

General Form of a Series in A.P.

𝑎, 𝑎+𝑑, 𝑎+2𝑑, 𝑎+3𝑑, ⋯, 𝑎+(𝑛−1)𝑑 let 𝑎=first term, 𝑑=common difference, 𝑙=last of 𝑛 terms, 𝑠=sum of 𝑛 terms; then 𝑙=𝑎+(𝑛−1)𝑑 𝑠=(𝑎+𝑙)𝑛2 𝑠={2𝑎+(𝑛−1)𝑑}𝑛2

Proof

By writing the series in reversed order, and adding both series together.

Arithmetric Mean

Arithmetric mean between 𝑎 and 𝑏 = 𝑎+𝑏2

Sources and References

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