THE BINOMIAL THEOREM 7 



As B is of the second degree in n, C will be of the third degree 

 in //, and its most general form will be b + en + dn z + hn 3 , where 

 b, c, d, and h are constants. 



Let C = b + en + dn z + hn 3 



when n = 3 C = 1 b + 3c + Qd + 27h - 1 . . (1) 



when n = 4 C = 4 b + 4c + Wd + 64ft = 4 . . (2) 



when n = 5 C = 10 b + 5c + 25d + I25h 10 .. (3) 



when n = 6 C = 20 6 + 6c + 36d + 216ft = 20 . . (4) 



subtracting (1) from (2) c + Id + 37ft = 3 . . (5) 



subtracting (2) from (3) c + 9d + 61ft = G . . (6) 



subtracting (3) from (4) c + lid + 91ft = 10 . . (7) 



subtracting (5) from (6) 2d + 24ft = 3 . . (8) 



subtracting (6) from (7) 2d + 30ft = 4 . . (9) 



subtracting (8) from (9) Qh = 1 h = 



from (8) 2d = 3 - 24ft d = -* 



iB 



from (5) c = 3 - 7d - 37ft c =\ 



o 



from (1) b = I - 3c - 9d - 27ft b = 



T,, ^ n 3 n 2 n 



ThuS = ---- + 



= (n 2 - 3n + 2) 



_ n(n l)(n - 2) 



6 



This result agrees with the anticipated result for C. Hence 

 (a; + a) n - a;" + Aaa:"- 1 + BaV- 2 + Ca 3 a;"- 3 4- DaV- 4 + . 



n 



where A = - 



2 1-2 



n 2 n(n l)(n 2) 



3 1-2- 3 



n - 3 n(n - l)(n - 2W 



1-2.3.4 



The expansion can also be written as 



