BOOK OF MATHEMATICAL PROBLEMS. 

 prove that 



S m = \nabc ... , 2, +l = [n + 1 abc . 

 and 12 S m+9 = \n + 2 abc... {2 2 (a 9 ) + 



If a be any other quantity, and if S r now denote 



then will ^ = S a = S a = ... = ,_! = 0, 



S n = \n abc . . . , 2S n+l = [n + l abc . . . (2a + a + b + c . . .), 

 and 12 S. +2 = n + 2 abc . . . {2 2 (a 2 ) + 3 2 (06) + 6a 2 (a) + 6a 2 }. 



XII. Summation of Series. 

 If w^ denote a certain function of n, and 



the summation of the series means expressing S n as a function of 

 n involving only a fixed number of terms. The usual artifice by 

 which this is effected consists in expressing u n as the difference of 

 two quantities, one of which is the same function of n as the 

 other is ofw 1, (U n ?/"_,). This being done, we have at 

 once 



fS.s(U t -U^ + (U t - U)+ ...... +(U,- 0._,) S U.- U,. 



Thus, if u n be the product of r consecutive terms of a given 

 A. P., beginning with the n th , 



