Engler — Figurate Numbers. 



51 



sum of the first n terms of the preceding series. In this 

 case the expressions for the sums become : 

 For (e), by (1) and (2) 



^ (n — l)n , 



1-2 



For (f), by (2) and (3) 



^ n ju-^l) (n—l) n {n-\-l) 



1:2 



For (g), by (3) and (4) 



1-2:3 



g ^ n{n+l){n+2) ^ {n—l)n(n-\-l){n+2) ^ 



l'2-3 

 For (h), by (4) and (5) 



n{n+l){n+2){n+3) 



1-2-3-4 



S = 



1-2-3-4 



+ 



{n—l)n (n+l) (t^+2)(^+3) 

 l-2-3t4-5 



d 



For d = o, these series become 



1 2 5 8 n 

 1 I 6 14 25" 

 1 6 20 45 

 1 1 5 25 70 



Table 22. 



For d=2. 



Tabic 23. 

 which is identical with Table 14. 



(12) 



(13; 



(14) 



(15) 



