FIGURATE NUMBERS. The nature of F. N. will be understood from the follow ing table: I, 2, 8, 4, 5, 6, 7, etc.
I. 1, 3, 6, 10, 15, 21, 28, etc.
II. 1, 4, 10, 20, 35, 56, 84, etc.
III. 1, 5, 15, 35, 70, 126, 210, etc. etc. etc.
The natural numbers are here taken as the basis, and the first order of F. N. is formed from the series by successive additions; thus, the 5th number of the first order is the sum of the first five natural numbers. The second order is then formed from the first in the same way; and so on.
If instead of the series of natural numbers, whose difference is 1, we take series whose differences are 2, 3, 4, etc., we may form as many different sets of figurate numbers. Thus: 1, 3, 5, 7, 9, etc. • I. 1, 4, 9, 16, 25, etc.
II. 1, 5, 14, 30, 55, etc.
III. 1, 6, 20, 50, 105, etc. etc. etc.
Or 1, 4, 7, 10, 13, etc.
I. 1, 5, 12, 22, 35, etc.
II. 1, 6, 18, 40, 75, etc.
III. 1, 7, 25, 65, 140, etc. etc. etc.
The name figurate is derived from the circumstance, that the simpler of them may oe represented by arrangements of equally distant points, forming geometrical figures.
The numbers belonging to the first order receives the general name of polygonal, and the special names of triangular, square, pentagonal, etc., according as the difference of the basis is 1, 2, 3, etc. Those of the second order are called pyramidal numbers, and according to the difference of the basis, are triagonally, quadragonally, or pentagonally pyramidal. The polygonal numbers may be represented by points on a surface; the pyramidal by piles of balls.
The general formula for polygonal numbers, from which any particular one may be found by substituting the proper values for n and r is, (r - 2) -(r- 4) n 2 where n= number of the term required, r = the denomination (3 if triagonal, 5 if pen tagonal, etc.).