Engler — Figurate Numbers. 



39 



If now we write a immber of rows of the series 

 1, 2, 2, 4 . . ., 



Table 2. 



in the form of a rectangle, draw the zigzag diagonal, and 

 divide the rectangle of numbers into two triangles as be- 

 fore, it becomes evident on inspection that the sum of the 

 numbers in each column of the right-hand triangle is 

 equal to twice the sum of the numbers in the correspond- 

 ing row of the left-hand triangle ; and that, therefore, the 

 sum of all the numbers in the left-hand triangle is equal 

 to one-half the sum of all the numbers in the right-hand 

 triangle, or equal to one-third the sum of all the numbers 

 in the whole rectangle. 



As before, the sum of all the numbers in the whole 

 rectangle of numbers is equal to the sum of all the num- 

 bers in each row multiplied by the number of rows. 



The sum of the numbers in each row of the rectangle, by- 



formula (2), if for n we put n-\-l, is 



{n-hl){n+2 ) 

 1-2 



the 



number of rows is n; therefore the sum of all the numbers 

 in the whole rectangle will be 



n{n-{-l){n^2) 



S 



1-2 



and the sum of all the numbers in the left-hand triangle 

 will be 



S = 



H(w-fl)(7i-f2) 



1-2-3 



(3) 



