374 Trans. Acad. Sci. of St. Louis. 
eal series of consecutive odd numbers, the first term of 
which is unity and the final term of which is 
I == Ia? —1 (1) 
and that the number of terms in the series is 
N=2 (2) 
The summation value is 
Htlyesor 
When » is an odd number, such a series will represent 
x if « 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 x8 and n—2, when x=—2 and 
= 6, or when x =4 and n=3. The values of / and N 
can also be computed and yield the same results for 64’, 
8*, 4°, 2”, and 16°. Table II, adjoining, represents the 
series for x = 2 and 3 for various values n. 
The result obtained by Nicomachus indicates that when 
nm is an odd number the first term of the series of consecu- 
tive odd numbers which is to have a sum 2® 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 
ati aes 
as +1 
Ret n—l 
l=a7* +a * =] (4) 
The number of terms in the series is | 
at. 
oie as ae (5) 
