35 ^ 



the property now stated may be otherwise proved in a very 

 simple manner immediately from the expression for the sum 

 of each series, I have also added the rules for filling up the 

 intervals of the terms, which Euler has not given ; and shew- 

 ed under what circumstances, other series will have the 

 same property. 



First, for the seines 1, 2, 4, 8, &c. 



The sum of 1+2+4+8+ 2 ""^ "~l-S; hence, 



8+1=2 the next term. The difference, then, between the 



sum 5' of w terms, and the next term 2," is 1 ; therefore the 

 sum S of n lerms, carries on within 1 of the next term. If 

 therefore you can for n terms, make up all the natura,! num- 

 bers to their sum, you make them all up to the number next 



less than the next term S ; and by adding all those numbers 



K 



to 2 , you get all the numbers to the number next less than 

 «+i 



2 . If therefore the rule be true for n terms, it must be 



true for n+l terms. Now if we take two terms 1, 2, we get 

 1+2=3, that is, we get all the numbers as far as the sum of 

 the two numbers, and within 1 of the next term. But, as pro- 

 ved above, if the rule be true for 2 terms, it must be true for 



3 terms; if true for 3 terms, it must be true for 4 terms ; 

 and so on; hence, the rule is true in general. 



