48 Trans, Acad, Sci. of St, Louis. 



The foregoing set of series of figurate numbers may be 

 regarded as a special case; but the method described may 

 be extended to more general cases as follows : 



A. If instead of Table 6 we write 



(a) 1 l-\-d 1 + d 1 + d 1 + d 



(b) 1 2 + d 3 + 2d 4: + Sd 5 + 4d 



(c) 1 3 + d 6 + 3cZ 10+6d 15+lOd 



(d) 1 4 + ^ 10+4c^ 20+10(^ 35 + 20d 



Table 13. 



where d is any integer, it will be seen that in this set of 

 series, the n^^ term of any series is equal to the sum of 

 the first n terms of the preceding series, and that the defi- 

 nition of figurate numbers is therefore satisfied. In this 

 set each of the rows may be separated into two series 

 whose sum is the sum of the corresponding series in the 

 table. 



Thus the series (a), (b), (c), (d) may be written 



(a) (1 1 1 1 1 . .) + (0 1 1 1 1 . .) <^ 



(b) (1 2 3 4 5 . .) + (0 1 2 3 4 . .) (i 



(c) (1 3 6 10 15 . .) + (0 1 3 6 10 . .) <^ 



(d) (1 4 10 20 35 . .) + (0 1 4 10 20 . .) d 



For the sum of (a) we have by (1) 



S = n^(n~l)d (8) 



For the sum of (b) we have by (2) 



n{n + l) {n—l)n 



1-2 1-2 



For the sum of (c) we have by (3) 



(9) 



n (n +1) (n+ 2) (n-1) n (n+l) ^ ,.^v 



