Various Tables of Annuities, j 323 



^3/0 + ^y»i + &c. + Ai^(.-,)„i =^„.„.- - 3/0. 



A>o+ A^'j/.^ + A^3/,„_,)„, = A-3/„„i - A^o, 



when Mi= 1, i = - 

 n 



the sum of the series is equal to 



n 

 1 



+ iji + 2 + 3 ^n-\\\y„,-y,\ 



1 .2M^ 



{l.>j-l+2.w-2+3.n-3 +W-1 .l}{A«/„,-AJ/p} 



+ 3 {1 .n — 1 .2n — 1+2.W— 2.2n-2.... 



The coefficient of A»2/,„ — A^yj is equal to the coefficient of 

 a;'-' in the development of 



(l+a;)"'-(l+x) 

 l-(l+a:)" ' 



or, in other words, if this coefficient be called z,, 



is the generating function of s,, and since ni = 1, 



(l+xY'-(l+.v) _ X 



l-(l+a;)' ~ (l + a;)'-l 



1 i—\ i— l.i+1 , I— l.i+1 , „ 



= -. T- X + -. :^ -. x^ + &c. 



I 21 121 24t 



n—l n—l.n+l „ n— l.n+l , . 



= « + X 7- X- + — ar + &c. 



2n 12n 24n 



SS2 



