94 BISHOP TERROT ON THE SUMS OF THE DIGITS OF NUMBERS. 



Prop. XIII. — Of Polygonal Nunibers. 



In any series of polygonal numbers, n the root of notation being even, the 

 sum of the digits of the (« + w— 1)* term = sum of s"" term. 



jjj 2 © — 'Vti — ^ (f 



For every polygonal number is of the form P= '—^ '-, where m is 



the number of the order, and s that of the term. 



For s substitute g + n—\. 



^ OT— 2 X (g+w— 1)^— (w— 4).(g + w— 1 ) 

 ^- 2 



■w— 2.*^— m — 4.S m— 2.(2 s.n— I + w— ll^) — ot— 4.w— 1 

 = 2 "^ 2 



= P + n-l . ^ 2 • 



But n being even, the fractional expression is integei', whether ??? be even or odd. 

 Therefore P and P' have same ultimate sum. (Prop. VI. Cor.) 



If n be odd, the fractional expression is integer only when m is even. 



The same inference might at once be drawn from the consideration, that the 

 s"' term of any order of polygonals is the sum of s tenns of an arithmetical series. 



Prop. XIV. 

 If, as in om- notation, n be even, the s*, (s+p.n^iy^ and (p.>T^l— «+!)"' terms 

 of a triangular series have all the same ultimate sum. 



s.s + 1 



In this case, m=3, and «"" term = ' 



2 



Therefore, (s+p.rr^y^ = ^'P^^\''~^ ^.n-l + '-^. Here the co-efficient 



of n^l is integer, whether^ be odd or even : and therefore sum of {s+p.n^lf^ 

 term = sum of s"" . 



Again, 



(p:^l -JTiy^ term = {P-n-T-s + l).{P-n-l-s) 



2 



_ j^.n— 1—2 sp+p -, *.« + ! 



2 .»-i + ^-. 



and as the coefficient of ra-1 is again integer, whether p be odd or even. Sum of 



(;j.«^T; -/+!)"'= sum of s'^ 



Ex. The triangular numbers are, — • 



1, 3, 6, 10, 15, 21, 28, 36, 45, 55, 66, 78, &c. 

 Sums, 1, 3, 6, 1, 6, 3, 1, 9, 9, 1, 3, 6, &c. 



