146 



Each of the following triangles is formed of a series of arithmeti- 

 cal progressions, the number of terms increasing successively by 

 unity. 



The first term of an arithmetical progression of n terms having a 

 common difference I, and whose sum is n a , is equal to 



1. SQUARE NUMBERS. 



If S=w 2 , the first term =n+i(l-n). 



M 



A. 



Every square is the sum of an arithmetical progression of n 

 terms, the first term of which is unity and the difference 2. 



1 =13 



1 + 3 = 2* 



1+3 + 5 =3 



1 + 3 + 5 + 7 =4* 



1+34.5 + 7 + 9 =59 



1 + 3 + 5 + 7 + 9+11 =6 2 



1 + 3 + 5+7 + 9+11 + 13 = 7 2 



Thus, every square number is formed by the addition of a series 

 of odd numbers commencing with unity; a result universally 

 known. 



The difference of any two squares is either ail odd number, or the 

 sum of consecutive odd numbers. 



Each series may be resolved into two others consisting of alter- 

 nate odd numbers, the respective sums of which are two adjacent 

 triangular numbers, the addition of which it is well known forms a 

 square. Ex. : 



1+5 + 9 + 13=28 

 3 + 7 + 11 = 21. 



49 =7 2 



