Algebra

Polygonal Numbers

The 𝑛^th term of the 𝑟^th order of polygonal numbers is equal to the sum of 𝑛 terms of an Arith. Prog. whose first term is unity and common difference 𝑟−2; that is =𝑛2{2+(𝑛−1)(𝑟−2)}=𝑛+12𝑛(𝑛−1)(𝑟−2) 287 The sum of 𝑛 terms =𝑛(𝑛+1)2+𝑛(𝑛−1)(𝑛+1)(𝑟−2)6 By resolving into two series. Order𝑛th terms 111111111 2𝑛1234567 312𝑛(𝑛+1)13610152128 4𝑛214916253649 512𝑛(3𝑛−1)151222355170 6(2𝑛−1)𝑛161528456691 ⋯⋯⋯⋯⋯⋯⋯⋯⋯ 𝑟𝑛+𝑛(𝑛−1)2(𝑟−2)1𝑟3+3(𝑟−2)4+6(𝑟−2)5+10(𝑟−2)6+15(𝑟−2)⋯ In practice, to form, for instance, the 6^th order of polygonal numbers- write the first three terms by the formula, and form the rest by the method of differences. Ex.  1 6 15 28 45 66 91 120 ⋯   5 9 13 17 21 25 29 ⋯ [𝑟−2=4]  4 4 4 4 4 4 ⋯ 288

Sources and References

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