Every cube n 3 is the sum of an arithmetical progression of n 

 terms, the first term of which is unity, and the difference 2( + l). 



1 =1 



1 + 7 =23 



1+9 + 17 =3 3 



1 + 11 + 21+31 =4 3 



1 + 13 + 25 + 37 + 49 =5 3 



1 + 15 + 29 + 43 + 57 + 71 = 6 3 



1 + 17+33 + 49 + 65 + 81 + 97 =7 3 



D. 



Every cube n 3 is the sum of an arithmetical progression of n 

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



1 = 1 3 



2 + 6 =2 3 



3 + 9 + 15 = 33 



4+12 + 20 + 28 =4 3 



5+15 + 25+35 + 45 =5 



6+18 + 30+42+54 + 66 =6 3 



7 + 21+35 + 49 + 63 + 77 + 91 . . . . =7 3 



The last terms of these series are the alternate triangular num- 

 bers. If they be respectively divided by the first terms, the quo- 

 tients will be the series of odd numbers. 



E. 



Every cube n 3 is the sum of an arithmetical progression of n 

 terms, the first term of which is (w 2 w+1), and the difference 2. 



1 =13 



3 + 5 =2' 



7 + 9 + 11 =33 



13 + 15 + 17 + 19 =4 3 



21 + 23 + 25 + 27 + 29 =5 3 



31+33 + 35 + 37 + 39 + 41 =6 3 



43+45 + 47 + 49 + 51+53 + 55 ,.=7 3 



