374 Trans. Acad. Set. of St. Louis. 



cal series of consecutive odd numbers, the first term of 

 which is unity and the final term of which is 



l = 2x 2 — 1 (1) 



and that the number of terms in the series is 



m 



N^x 2 (2) 



The summation value is 



1±1n = x» = N 2 (3) 



"When n is an odd number, such a series will represent 

 of- if x is the square of some integral. The sum of a con- 

 secutive series of odd numbers from 1 to 15 is 64. The 

 final term and the number of terms satisfy equations (1) 

 and (2) above when x = 8 and w = 2, when # — 2 and 

 n = 6, or when x — 4 and n = 3. The values of I and N 

 can also be computed and yield the same results for 64 2 , 

 8 4 , 4 6 , 2 12 , and 16 s . Table II, adjoining, represents the 

 series for x = 2 and 3 for various values n. 



The result obtained by Nicomachus indicates that when 

 n is an odd number the first term of the series of consecu- 

 tive odd numbers which is to have a sum x* should be 

 greater than unity. The results in Table III were readily 

 obtained by an inspection of the column of odd numbers 

 and the summation of various groups of consecutive 

 terms. The values of the first and last terms of the 

 series were found to be 



a+l n — 1 



a = x 2 — x i -j- 1 



n+l n — 1 



l = X 2+^2—1 (4) 



The number of terms in the series is 



n — 1 



N = x 2 ' (5) 



