58 



necessarily be 1,2, or 1, S, or the interval between the two 

 first terms cannot be filled up. The series must also have 

 this further property, that the sum preceding any term (P) 

 must reach at least half way from (P) to the next term (Q), 

 or to (Q — i). The following series have this property. 



1, 2, o, 10, 17, &c. 



1, 2, 7 17, 33, &c. 



1, 3, 6, 10, 15, &c. 



1, 3, 9, 19, 33, &c. 

 and many others ; but the series which requires the smallest 

 number of terms to fill up the interval from 1 to any given 

 number, is, 1, S, 9, 27, 81, &c. 



