198 Mr. Blackburn's ^w«/?///c«/ Theorevis 



where the index n shows that there are n factors, and the 

 index 2 that the indices of each succeeding factor are double 

 of the preceding. 



(2.) Hence it appears that if an odd number of terms of 

 a geometric series be raised to any power of two, the sum of 

 the terms so raised is divisible by the original series. Also 

 that the quotient consists of a number of factors, each of 

 which is a geometric series of the same number of terms as 

 the original one, having the signs of the alternate terms ne- 

 gative, the number of series or factors being denoted by the 

 index of two. The first factor is the original series with the 

 signs of the alternate terms changed, and the indices of each 

 succeeding factor obtained, by doubling those of the pre- 

 ceding. 



Example, 



Let w = 1, then by the formula 



1 I 3 ■ 4 . (m— 1)2 



— ^ — = 1— r + 7-+ r 



l4-r+r^+ 7-^-1 



or, jl+r + r^+ r^^-^} .{i-r-fr^- i^'-^\ 



= l+r^ + r'*+ r^»*-^)2 



from which it appears that if an odd number of terms of a 

 geometric series, be multiplied by the same terms with the 

 signs of the alternate terms changed ; the product will be the 

 sum of the squares of the original series. 



(% \ Cor ^^ ^ -^ "• ^ 



(J.; car. ^^ (m-i)/c 



1 +r +r + r 



= |1— r +r — ^ r^ ' j . 



(4.) Cor, 2. Every algebraical expression of the form 



l-fr^ _|.,.2-2 +... /"»-!) 2 Ij. divisible into ;;+l factors, 

 each of which is a geometric series of m terms, and y; of 

 which factors have the signs of the even terms negative, thus 



1+^2 + ;. 2.2'; ^(».-l)2''= |)+;.+,.2+ ...r"-'} X 



^X-r + r"^- ,.'»-' J- X 



