Engler — Figurate Numbers. 



51 



sum of the first n terms of the preceding series, 

 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 (n-i-l) jn—l) n {n-j-l) 



1-2 



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



1-2-3 



^ n{n+l)in+2) {n—l)n{n+l){n+2) 

 1-2-3 l-2r3r4 



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



^__ n{n+l){n+2){n+3) _^ 



1-2-3-4 



{tv—l)}}, (»+!) (»^+2)(^+3) 

 l-2:3r4-5 



d 



For d = o, these series become 



1 2 5 8 II 



1 I 6 l4 25^ 



1 6 20 45 



1 1 5 25 70 



Table 22. 



For ^=2, 



Tabic 23. 

 which is identical with Table 14, 



In this 



(12) 



(13j 



(14) 



(15) 



