56 



Secondly, for the Series 1 , 3, 9, 27, &c. 



The sum of 1+3 + 9+27+ s'~L^zzS ; hence, 



tl 



25+1=3 the next term. The difference, then, between the sum 

 5 of w terms and the next term 3 is S+i ; therefore the sun» 



S subtracted from the next term 3", leaves 5+ 1] that is, it 

 brings jou back to the number next greater than the sum S. 

 If therefore j-ou can for 7i terms make up all the numbers to 



S, the same numbers subtracted from the n+i" term will 

 bring you back to S+i, the number next greater than ■S'; 



thus you fill up all the numbers in the interval between the n 

 term and the w + l' term ; and if the same numbers be added 

 to the w-|-l" term, you make up all the numbers as far as the 

 sum of w+1 terms ; if therefore the rule be true for n terms, 

 it must be true for n+i terms. Now if we take two terms 

 1, 3, we have 3 — 1=2, 3+1=4, and 4 subtracted from the 

 next term 9> leaves 5 the next number greater than the sum , 

 of two terms. But, as proved 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. 



The intervals of the first series may be filled up by the 

 following Rule. 



o 



Let A be any number, and 2" the term next less than A. 

 Take ^^ next less than A—2"; 2* next less than ^—2"— 2'; 



